Kant and the Prolegomena to Any Future Metaphysics

According to Kant metaphysics is the â€Å"occupation of reason with itself†. In more concrete terms, it is the mind making logical connections between a priori concepts and coming to an objective truth thereby, without reference to experience.[1] The question posed by him in the Prolegomena is whether such an objective truth is at all possible. The conclusion derived in the end is that there is indeed such an objective truth, which is effected through pure reason. But equally important in the assertion is that such metaphysics is beyond human understanding.The title â€Å"Prolegomena to Any Future Metaphysics† suggests that Kant does indeed anticipate an irrefutable metaphysics to be in the grasp of men in the future, but he never makes such a claim in the text itself. The thing that Kant aimed for was clarity in the field of metaphysical endeavor, and this is the ‘future metaphysics’. ‘Future’ can be interpreted in two ways here. First in the sense already suggested, so that metaphysical thinking is founded on a scientific basis, in which the terms and strategies it employs are well defined. But it can also be hinting at transcendental possibility, that by which all contradictions are resolved through â€Å"pure reason†.Scientific clarity is the aim, and thus Kant justifies the labor involved in Critique Of Pure Reason (1781), of which the Prolegomena was a sequel meant to make more accessible. He is at pains to point out that there is a moral obligation involved here. People cannot surrender themselves to unreason, because reason is the very make-up of the human, so postulates Kant. The suggestion that reason be abandoned was made by David Hume, who had spelt out a comprehensive theory of empirical skepticism.All our knowledge is through sense perceptions, therefore are entirely subjective, and cannot be tied into an absolute whole through the application of reason. It is merely by the means of custom that we ac quire a coherent worldview, he maintained.[2] Kant saw this as a capitulation to unreason. It was not just Hume’s personal viewpoint that mattered. It was indeed a wider crisis in metaphysics that he was addressing.When Newton’s physics could not be subsumed under any metaphysics, this engendered an intellectual confusion, and Hume’s solution was that metaphysics be abandoned as impossible. Kant enjoined that it is impossible to abandon metaphysics, for man reasons by necessity. Instead of finality we must aim for metaphysical clarity, and this is absolutely contingent upon us, indeed a moral obligation. He made what seem to be boastful claims about the crucial importance of the Critique in the history of metaphysics, but a closer examination will show that it is not from conceit, but rather from moral outrage.The true nature of metaphysics is laid out with scientific clarity in the Critique and the Prolegomena, and this is the essence that Kant wants to convey, not the final outcome. He takes Hume to task in the very opening of the Critique:  Although all our knowledge begins with experience, it does not follow that it arises entirely from experience. For it is quite possible that our empirical knowledge is a compound of that which we receive through impressions and that which our own faculty of knowing (incited by impressions) supplies from itself.[3]When considering sensual perception he first makes the distinction between a priori and a posteriori, the first suggesting an innate facility of the mind, and the second is a facility borne after the event. The second distinction is between analytical and synthetic propositions. In an analytic proposition the predicate is contained within the subject, such as â€Å"the flamingo is a bird†. In a synthetic proposition the predicate adds something new to the subject, such as â€Å"the flamingo is pink†. The pinkness is not in the definition of flamingo, but rather has to be got from observation, and therefore it is also a posteriori.Synthetic a posteriori propositions are employed in the field of natural science. On the other hand all mathematical truths are innate, i.e. we ascertain their truth before sensory perception. They are also synthetic: when we say â€Å"3+4=7†, then ‘7’ is a new concept, not contained in either ‘3’ of ‘4’. Mathematics holds the key to metaphysics, according to Kant. It demonstrates that synthetic a priori propositions are possible, which is contrary to normal expectation. We feel that whatever is innate is necessarily analytical. We are what we are, separated from the objective natural world beyond us. Against this instinctive point of view, Kant contended that we are not passive observers of an external world separated from us, but that with our innate faculties we â€Å"synthesize† our own subjective reality.The first stage of this synthesis is when we intuit objects in our perception. â€Å"Things in themselves† can never appear to us; we only have subjective sensory data to work with. It is a meaningless jumble of light, sound, touch, taste and smell, but then our faculty of sensibility intervenes and creates order out of this chaos. This faculty is synthetic a priori, and makes use of pure intuitions. Space is one such pure intuition. Newton had maintained that space is an external, absolute and inviolable reality. Kant counters that, no, space is pure intuition. Time is another such.Through the faculties of sensibility we come to make a judgment of perception. Thus far it is an entirely subjective viewpoint, with no objective framework to relate to that would link our views with those of others. This is the function of our judgment of experience. It too is synthetic a priori, and links the objects of perception into a rational order that facilitates understanding. This is done through pure concepts of understanding, and causation is one of t hem. Through this faculty we know that one event is cause to another, and thus wise we have come across Hume’s impasse, where he could find no rational construct that could link a effect to a cause when confined to empirical sense data.[4] â€Å"Cause and effect† is thus a concept of human understanding. Such understanding is composed of components that are a priori and synthetic, and it is meant to make the world intelligible to us.Just because the world is made intelligible, it does not imply that we do not meet contradiction. When we think we do so discursively, i.e. we think by making propositions in terms of subjects and predicates. But each subject we introduce is the predicate of another subject in an infinite chain. Because the absolute subject is beyond our grasp, discursive reason naturally leads to fallacies. In fact each truthful proposition will be found to have an equally valid refutation, which together are described as pairs of antimonies. Kant cites fo ur cosmological antimonies, one of which places infinite space against a limited one. He goes on to show that there is no contradiction in essence. As originating in the judgment of perception space does indeed have a beginning. But as regards human understanding space is necessarily infinite. The conflict arises from metaphysics failing to distinguish the noumenal (thing in itself) from the phenomenal (as appears to human understanding).Thus far does metaphysics gain clarity, but not finality. The human mind cannot help ponder on the questions of metaphysics, but it must come to terms with the fact that it is ‘bounded’. Human understanding is meant to make the outer world intelligible, and thus proves inadequate when the focus is redirected to the inner essence of the mind, which is the object that metaphysics must study. But the overriding lesson of metaphysics is that pure reason subsumes all. One must not despair of human reason, for one must know that it originates in pure reason and is overcome by it in the end.Subservience to pure reason is indeed a moral obligation. Other than clarity in metaphysics, which is not suitable for all, Kant advanced his categorical imperative: â€Å"I ought never to act except in such a way that I could also will that my maxim should become a universal law.†[5] We cannot help noticing that this is only a rewording of the golden rule of Christianity: â€Å"Do unto others as you would they should do unto you.†[6] Thus through clarity in metaphysics Kant can be said to have arrived at religious doctrine too.ReferencesHume, David. An Enquiry Concerning Human Understanding. Ed. Eric Steinberg. Boston: Hackett Publishing, 1993.Jeffrey, David L. A Dictionary of Biblical Tradition in English Literature.  Grand Rapids, MI: Wm. B. Eerdmans Publishing, 1992.Kant, Immanuel. Critique of Pure Reason. Trans. Werner S. Pluhar. Ed. Eric Watkins. Boston: Hackett Publishing Company, 1999.Kant, Immanuel. Groundwork of the Metaphysics of Morals. Ed. Mary Gregor. Cambridge: Cambridge University Press, 1998.Kant, Immanuel. Kant's Prolegomena to Any Future Metaphysics. Whitefish, MT: Kessinger Publishing, 2005.[1] Immanuel Kant, Kant's Prolegomena to Any Future Metaphysics, Whitefish, MT: Kessinger Publishing, 2005, p. 92.[2] David Hume, An Enquiry Concerning Human Understanding, Ed. Eric Steinberg, Boston: Hackett Publishing, 1993, p. 29.[3] Immanuel Kant, Critique of Pure Reason. Trans. Werner S. Pluhar, Ed. Eric Watkins, Boston: Hackett Publishing Company, 1999, p. 1. [4] Hume, Enquiry, p. 49. [5] Immanuel Kant, Groundwork of the Metaphysics of Morals, Ed. Mary Gregor, Cambridge: Cambridge University Press, 1998, p. 16. [6] David L. Jeffrey, A Dictionary of Biblical Tradition in English Literature,  Grand Rapids, MI: Wm. B. Eerdmans Publishing, 1992, p. 314.

The Relationship Between Shelia and Mrs. Birling in ‘an Inspector Calls’. Essay Essay Example

The Relationship Between Shelia and Mrs. Birling in ‘an Inspector Calls’. Essay Essay Example The Relationship Between Shelia and Mrs. Birling in ‘an Inspector Calls’. Essay Essay The Relationship Between Shelia and Mrs. Birling in ‘an Inspector Calls’. Essay Essay Research how Priestly presents and develops the relationship between Shelia and Mrs. Birling in ‘An Inspector Calls’ . Priestly uses duologue and phase waies to demo a nothingness between different coevalss. Although Mrs. Birling is Shelia’s female parent she still refers to her as a ‘girl [ s ] ’ . The audience can feel Mrs. Birling’s condescending tone and how she still retains the position that Shelia is an immature and nescient ‘girl [ s ] ’ even though she is a to the full grown grownup who will shortly be wed. This is dry as it is in fact Mrs. Birling who is the nescient character in the drama when she states that Eric and Shelia are ‘over-tired’ and will ‘be as diverted as’ the older Birlings are in the forenoon. but the audience know this is false and the younger coevals of Birlings have matured and learnt far more than their seniors. Priestly genuinely shows Shelia’s development and adulthood as she changes her address term for Mrs. Birling from ‘mommy’ to ‘mother’ . The usage of the word ‘mother’ is much more degage and formal and suggests Shelia has broken out of the conformance of her mother’s denigration sentiments. In this short infinite of text Priestly reveals Shelia as an adaptable and developing character in the drama. To both a modern twenty-four hours and 1945 audience. naming an engaged adult females a ‘girl’ is pathetic and patronizing. but so once more Shelia is a spoiled ‘mummy’s’ miss to get down with. Priestly is demoing how the younger coevals must lift up and take their topographic point in society alternatively of staying spoon Federal and ignorant. It takes a large event. Eva Smith’s decease. for Shelia to gain this but to a 1945 audience this would vibrate as with the wake of a war in forepart of them and ‘millions of Eva Smiths and John Smiths’ who died in that really war. priestly portrays that this is a better clip than any to take the reins. This suggests. to promote societal alteration. Priestly believes people need to be able to accommodate and alter their thoughts like the younger coevals in the text. Priestly generates tenseness and struggle between Shelia and Mrs. Birling by making Irony. When Shelia says ‘what do you intend by stating that? You talk as if we were responsible’ . this will subsequently go dry as Shelia is the character that focuses on and learns the most from the fact the household ‘were responsible’ . This denotes a self-conflict and development in Shelia’s character as she breaks off from her mother’s positions and becomes ‘responsible’ for her actions. Here Priestly is demoing the audience how we should look at ourselves instead than fault person else or disregard the effects as Mrs. Birling does. The divide between socialist and capitalist positions is emphasised by the apposition of ‘you’ and ‘we’ . the reiterates the older Birlings’ introverted doctrine and their denial to populate in a wider community but alternatively their desire ‘for a adult male to mind his ain business’ and go on to deny duty. Priestly uses duologue to show and develop dependence in the relationship between Shelia and Mrs. Birling. When Sheila is warning her female parent about ‘building up a wall’ between her and Eva. her following utterance high spots Shelia’s consciousness to the Inspector’s methods and intent. ‘He hasn’t started on you yet. This indicative mood is written in italics which in a drama would let an histrion to redact their tone and portray Priestly’s thoughts on coevals divide to the audience and emphasises how Shelia wants to protect her female parent from uncovering her egotistic and narrow minded positions. this allows the audience to place with her and see her as an surrogate hero in the drama. Here Priestly is demoing how the younger coevals are much more mature than many people think but lack the assurance to force through their thoughts and need the blessing and support of the older coevalss. With the war in 1945 societal alteration was inevitable as immature work forces and adult females were forced to function their state and take up their topographic point in society. throughout the drama priestly praises and encourages the young person of 1945. particularly adult females as they have a much more active function in society. through the characters of Shelia and Eric whom besides begin to take a base and gain their. potency and their value in society. Again Priestley’s duologue between Shelia and Mrs. Birling reveals tenseness at the bosom of their relationship. Despite being ‘very pleased with themselves’ the duologue and phase waies in the early pages of the drama suggest otherwise. When Shelia accuses Eric of being ‘squiffy’ Mrs. Birling responds. mentioning to her as a ‘girl [ s ] ’ . This conveys and immediate difference between the two females positions on societal etiquette. The exclamatory sentences Mrs. Birling uses reflect her daze and repulsive force that Shelia would utilize a conversational term such as ‘squiffy’ . The audience besides acquire an indicant that Mrs. Birling doesn’t regard Shelia as a adult adult female whose sentiments are valued. Shelia is a bride to be and is an grownup in her ain right nevertheless her female parent persists to utilize the instead take downing noun ‘ girl’ to depict her girl. Priestly is assailing the older coevals for compressing their children’s singularity and trying to make ringers of what he sees as a failing coevals. He expresses his belief that the authoritative phrase ‘children should be seen and non heard’ is exhaustively outdated and kids should be given a voice of their ain and be single.

Case Study Make Green Delicious Essay Example

Case Study Make Green Delicious Essay Example Case Study Make Green Delicious Essay Case Study Make Green Delicious Essay - A. - Introduction/Statement of the Problem: As the executive director of the National Hockey League Players’ Association (NHLPA), Bob Goodenow is faced with the issue of developing a communications strategy for the NHLPA. It must establish a strong bargaining stance for the players that effectively communicate their position on the salary cap issue, while also retain the loyalty of hockey fans. B. - Analysis: The current Collective Bargaining Agreement includes a limited salary cap and entry-level salary cap, and expires September 15th, 2004. It is no longer successful at controlling rapid growth in player salaries and forming a direct link between salary growth and league revenues. The Unified Report of Operations claims that the majority of NHL teams have lost money in recent years. Owners believe that there are â€Å"major deficiencies in the current contract that has led to significant financial losses and a competitive imbalance. Smaller teams can’t compete with larger teams; therefore the league can’t be competitive. There is a transition from a local market to a league-wide market, so players’ salaries are escalating to levels that are unsustainable. Entry-level salary caps are ineffective due to signing bonuses. The NHL revenue growth equals 173%, while the players’ salary growth equals 261% over the term of the CBA. The players’ salaries account for 75% of NHL revenues, which is much higher than other sports. Owners believe a mechanism is needed to control player salaries and tie them to revenues. Players question the validity of the financial information, URO, and Levitt Report. Players argue against the proposed solutions and say free market forces should determine players’ salaries. There is no validity in the owners’ financial statements, which may support that the lack of revenues is due to poor business decisions, not high player salaries. C. - Alternatives: A) Strategy that focuses on the players’ current stance on the salary cap, using free market forces, questionable financial validity, and poor business decisions as their platforms for gaining the media and the fans support. B) Strategy that emphasizes on the lack of financial transparency of the NHL and its owners that has led a delayed agreement. C) Continue with the current communications strategy. D. - Recommendation: Bob is recommended to choose alternative A and pursue a communications strategy, which emphasizes the NHL players’ current position on the proposed salary cap. E. - Implementation Plan: Bob should begin by clearly establishing the players’ values and goals in regards to the negotiation and how he intends on communicating them to reach the fans. This would involve taking the players’ stance on the salary cap and communicating it consistently through the NHLPA’s website, players’ personal social media, radio interviews, etc. It would include reasoning to support the players’ position, such as the lack of transparency and validity in financial information presented by the NHL owners, as well as the importance of free markets in the determination of salaries. By using the players as a source of free media and publicity, the NHLPA can emphasize their position and sympathize with the fans and their common love for the game. Here, the players can explain that a salary cap would essentially be equal to a salary cut for players. This is something that nobody would agree with if it came to their own job, whether they are rich or poor. By effectively communicating the players’ stance on the proposed salary cap to the fans and public, the NHLPA can gain a favourable bargaining position over the NHL owners for the upcoming negotiations in September 2004. It will also help retain fan loyalty to the NHL and its organizations, which is a common interest for both parties.

The 13 Hardest SAT Math Questions Ever

The 13 Hardest SAT Math Questions Ever SAT / ACT Prep Online Guides and Tips Want to test yourself against the most difficult SAT math questions? Want to know what makes these questions so difficult and how best to solve them? If you’re ready to really sink your teeth into the SAT math section and have your sights set on that perfect score, then this is the guide for you. We’ve put together what we believe to be the 13 most difficult questions for the new 2016 SAT, with strategies and answer explanations for each. These are all hard SAT Math questions from College Board SAT practice tests, which means understanding them is one of the best ways to study for those of you aiming for perfection. Image: Sonia Sevilla/Wikimedia Brief Overview of SAT Math The third and fourth sections of the SAT will always be math sections. The first math subsection (labeled "3") does not allow you to use a calculator, while the second math subsection (labeled as "4") does allow the use of a calculator. Don't worry too much about the no-calculator section, though: if you're not allowed to use a calculator on a question, it means you don’t need a calculator to answer it. Each math subsection is arranged in order of ascending difficulty (where the longer it takes to solve a problem and the fewer people who answer it correctly, the more difficult it is). On each subsection, question 1 will be â€Å"easy† and question 15 will be considered â€Å"difficult.† However, the ascending difficulty resets from easy to hard on the grid-ins. Hence, multiple choice questions are arranged in increasing difficulty (questions 1 and 2 will be the easiest, questions 14 and 15 will be the hardest), but the difficulty level resets for the grid-in section (meaning questions 16 and 17 will again be â€Å"easy† and questions 19 and 20 will be very difficult). With very few exceptions, then, the most difficult SAT math problems will be clustered at the end of the multiple choice segments or the second half of the grid-in questions. In addition to their placement on the test, though, these questions also share a few other commonalities. In a minute, we'll look at example questions and how to solve them, then analyze them to figure out what these types of questions have in common. But First: Should You Be Focusing on the Hardest Math Questions Right Now? If you’re just getting started in your study prep (or if you’ve simply skipped this first, crucial step), definitely stop and take a full practice test to gauge your current scoring level. Check out our guide to all the free SAT practice tests available online and then sit down to take a test all at once. The absolute best way to assess your current level is to simply take the SAT practice test as if it were real, keeping strict timing and working straight through with only the allowed breaks (we know- probably not your favorite way to spend a Saturday). Once you’ve got a good idea of your current level and percentile ranking, you can set milestones and goals for your ultimate SAT Math score. If you’re currently scoring in the 200-400 or the 400-600 range on SAT Math, your best bet is first to check out our guide to improving your math score to be consistently at or over a 600 before you start in trying to tackle the most difficult math problems on the test. If, however, you're already scoring above a 600 on the Math section and want to test your mettle for the real SAT, then definitely proceed to the rest of this guide. If you’re aiming for perfect (or close to), then you’ll need to know what the most difficult SAT math questions look like and how to solve them. And luckily, that’s exactly what we’ll do. WARNING: Since there are a limited number of official SAT practice tests, you may want to wait to read this article until you've attempted all or most of the first four official practice tests (since the questions below were taken from those tests). If you're worried about spoiling those tests, stop reading this guide now; come back and read it when you've completed them. Now let's get to our list of questions (whoo)! Image: Niytx/DeviantArt The 13 Hardest SAT Math Questions Now that you’re sure you should be attempting these questions, let’s dive right in! We've curated 13 of the most difficult SAT Math questions for you to try below, along with walkthroughs of how to get the answer (if you're stumped). No Calculator SAT Math Questions Question 1 $$C=5/9(F-32)$$ The equation above shows how temperature $F$, measured in degrees Fahrenheit, relates to a temperature $C$, measured in degrees Celsius. Based on the equation, which of the following must be true? A temperature increase of 1 degree Fahrenheit is equivalent to a temperature increase of $5/9$ degree Celsius. A temperature increase of 1 degree Celsius is equivalent to a temperature increase of 1.8 degrees Fahrenheit. A temperature increase of $5/9$ degree Fahrenheit is equivalent to a temperature increase of 1 degree Celsius. A) I onlyB) II onlyC) III onlyD) I and II only ANSWER EXPLANATION:Think of the equation as an equation for a line $$y=mx+b$$ where in this case $$C= {5}/{9} (F−32)$$ or $$C={5}/{9}F −{5}/{9}(32)$$ You can see the slope of the graph is ${5}/{9}$, which means that for an increase of 1 degree Fahrenheit, the increase is ${5}/{9}$ of 1 degree Celsius. $$C= {5}/{9} (F)$$ $$C= {5}/{9} (1)= {5}/{9}$$ Therefore, statement I is true. This is the equivalent to saying that an increase of 1 degree Celsius is equal to an increase of ${9}/{5}$ degrees Fahrenheit. $$C= {5}/{9} (F)$$ $$1= {5}/{9} (F)$$ $$(F)={9}/{5}$$ Since ${9}/{5}$ = 1.8, statement II is true. The only answer that has both statement I and statement II as true is D, but if you have time and want to be absolutely thorough, you can also check to see if statement III (an increase of ${5}/{9}$ degree Fahrenheit is equal to a temperature increase of 1 degree Celsius) is true: $$C= {5}/{9} (F)$$ $$C= {5}/{9} ({5}/{9})$$ $$C= {25} /{81} (\which \is ≠  1)$$ An increase of $5/9$ degree Fahrenheit leads to an increase of ${25}/{81}$, not 1 degree, Celsius, and so Statement III is not true. The final answer is D. Question 2 The equation ${24x^2 + 25x -47}/{ax-2} = -8x-3-{53/{ax-2}}$ is true for all values of $x≠ 2/a$, where $a$ is a constant. What is the value of $a$? A) -16B) -3C) 3D) 16 ANSWER EXPLANATION: There are two ways to solve this question. The faster way is to multiply each side of the given equation by $ax-2$ (so you can get rid of the fraction). When you multiply each side by $ax-2$, you should have: $$24x^2 + 25x - 47 = (-8x-3)(ax-2) - 53$$ You should then multiply $(-8x-3)$ and $(ax-2)$ using FOIL. $$24x^2 + 25x - 47 = -8ax^2 - 3ax +16x + 6 - 53$$ Then, reduce on the right side of the equation $$24x^2 + 25x - 47 = -8ax^2 - 3ax +16x - 47$$ Since the coefficients of the $x^2$-term have to be equal on both sides of the equation, $−8a = 24$, or $a = −3$. The other option which is longer and more tedious is to attempt to plug in all of the answer choices for a and see which answer choice makes both sides of the equation equal. Again, this is the longer option, and I do not recommend it for the actual SAT as it will waste too much time. The final answer is B. Question 3 If $3x-y = 12$, what is the value of ${8^x}/{2^y}$? A) $2^{12}$B) $4^4$C) $8^2$D) The value cannot be determined from the information given. ANSWER EXPLANATION: One approach is to express $${8^x}/{2^y}$$ so that the numerator and denominator are expressed with the same base. Since 2 and 8 are both powers of 2, substituting $2^3$ for 8 in the numerator of ${8^x}/{2^y}$ gives $${(2^3)^x}/{2^y}$$ which can be rewritten $${2^3x}/{2^y}$$ Since the numerator and denominator of have a common base, this expression can be rewritten as $2^(3x−y)$. In the question, it states that $3x − y = 12$, so one can substitute 12 for the exponent, $3x − y$, which means that $${8^x}/{2^y}= 2^12$$ The final answer is A. Question 4 $${8-i}/{3-2i}$$ If the expression above is rewritten in the form $a+bi$, where $a$ and $b$ are real numbers, what is the value of $a$? (Note: $i=√{-1}$) ANSWER EXPLANATION: To rewrite ${8-i}/{3-2i}$ in the standard form $a + bi$, you need to multiply the numerator and denominator of ${8-i}/{3-2i}$ by the conjugate, $3 + 2i$. This equals $$({8-i}/{3-2i})({3+2i}/{3+2i})={24+16i-3+(-i)(2i)}/{(3^2)-(2i)^2}$$ Since $i^2=-1$, this last fraction can be reduced simplified to $$ {24+16i-3i+2}/{9-(-4)}={26+13i}/{13}$$ which simplifies further to $2 + i$. Therefore, when${8-i}/{3-2i}$ is rewritten in the standard form a + bi, the value of a is 2. The final answer is A. Question 5 In triangle $ABC$, the measure of $∠ B$ is 90 °, $BC=16$, and $AC$=20. Triangle $DEF$ is similar to triangle $ABC$, where vertices $D$, $E$, and $F$ correspond to vertices $A$, $B$, and $C$, respectively, and each side of triangle $DEF$ is $1/3$ the length of the corresponding side of triangle $ABC$. What is the value of $sinF$? ANSWER EXPLANATION: Triangle ABC is a right triangle with its right angle at B. Therefore, $\ov {AC}$ is the hypotenuse of right triangle ABC, and $\ov {AB}$ and $\ov {BC}$ are the legs of right triangle ABC. According to the Pythagorean theorem, $$AB =√{20^2-16^2}=√{400-256}=√{144}=12$$ Since triangle DEF is similar to triangle ABC, with vertex F corresponding to vertex C, the measure of $\angle ∠  {F}$ equals the measure of $\angle ∠  {C}$. Therefore, $sin F = sin C$. From the side lengths of triangle ABC, $$sinF ={\opposite \side}/{\hypotenuse}={AB}/{AC}={12}/{20}={3}/{5}$$ Therefore, $sinF ={3}/{5}$. The final answer is ${3}/{5}$ or 0.6. Calculator-Allowed SAT Math Questions Question 6 The incomplete table above summarizes the number of left-handed students and right-handed students by gender for the eighth grade students at Keisel Middle School. There are 5 times as many right-handed female students as there are left-handed female students, and there are 9 times as many right-handed male students as there are left-handed male students. if there is a total of 18 left-handed students and 122 right-handed students in the school, which of the following is closest to the probability that a right-handed student selected at random is female? (Note: Assume that none of the eighth-grade students are both right-handed and left-handed.) A) 0.410B) 0.357C) 0.333D) 0.250 ANSWER EXPLANATION: In order to solve this problem, you should create two equations using two variables ($x$ and $y$) and the information you’re given. Let $x$ be the number of left-handed female students and let $y$ be the number of left-handed male students. Using the information given in the problem, the number of right-handed female students will be $5x$ and the number of right-handed male students will be $9y$. Since the total number of left-handed students is 18 and the total number of right-handed students is 122, the system of equations below must be true: $$x + y = 18$$ $$5x + 9y = 122$$ When you solve this system of equations, you get $x = 10$ and $y = 8$. Thus, 5*10, or 50, of the 122 right-handed students are female. Therefore, the probability that a right-handed student selected at random is female is ${50}/{122}$, which to the nearest thousandth is 0.410. The final answer is A. Questions 7 8 Use the following information for both question 7 and question 8. If shoppers enter a store at an average rate of $r$ shoppers per minute and each stays in the store for average time of $T$ minutes, the average number of shoppers in the store, $N$, at any one time is given by the formula $N=rT$. This relationship is known as Little's law. The owner of the Good Deals Store estimates that during business hours, an average of 3 shoppers per minute enter the store and that each of them stays an average of 15 minutes. The store owner uses Little's law to estimate that there are 45 shoppers in the store at any time. Question 7 Little's law can be applied to any part of the store, such as a particular department or the checkout lines. The store owner determines that, during business hours, approximately 84 shoppers per hour make a purchase and each of these shoppers spend an average of 5 minutes in the checkout line. At any time during business hours, about how many shoppers, on average, are waiting in the checkout line to make a purchase at the Good Deals Store? ANSWER EXPLANATION: Since the question states that Little’s law can be applied to any single part of the store (for example, just the checkout line), then the average number of shoppers, $N$, in the checkout line at any time is $N = rT$, where $r$ is the number of shoppers entering the checkout line per minute and $T$ is the average number of minutes each shopper spends in the checkout line. Since 84 shoppers per hour make a purchase, 84 shoppers per hour enter the checkout line. However, this needs to be converted to the number of shoppers per minute (in order to be used with $T = 5$). Since there are 60 minutes in one hour, the rate is ${84 \shoppers \per \hour}/{60 \minutes} = 1.4$ shoppers per minute. Using the given formula with $r = 1.4$ and $T = 5$ yields $$N = rt = (1.4)(5) = 7$$ Therefore, the average number of shoppers, $N$, in the checkout line at any time during business hours is 7. The final answer is 7. Question 8 The owner of the Good Deals Store opens a new store across town. For the new store, the owner estimates that, during business hours, an average of 90 shoppers per hour enter the store and each of them stays an average of 12 minutes. The average number of shoppers in the new store at any time is what percent less than the average number of shoppers in the original store at any time? (Note: Ignore the percent symbol when entering your answer. For example, if the answer is 42.1%, enter 42.1) ANSWER EXPLANATION: According to the original information given, the estimated average number of shoppers in the original store at any time (N) is 45. In the question, it states that, in the new store, the manager estimates that an average of 90 shoppers per hour (60 minutes) enter the store, which is equivalent to 1.5 shoppers per minute (r). The manager also estimates that each shopper stays in the store for an average of 12 minutes (T). Thus, by Little’s law, there are, on average, $N = rT = (1.5)(12) = 18$ shoppers in the new store at any time. This is $${45-18}/{45} x 100 = 60$$ percent less than the average number of shoppers in the original store at any time. The final answer is 60. Question 9 A grain silo is built from two right circular cones and a right circular cylinder with internal measurements represented by the figure above. Of the following, which is closest to the volume of the grain silo, in cubic feet? A) 261.8B) 785.4C) 916.3D) 1047.2 ANSWER EXPLANATION: The volume of the grain silo can be found by adding the volumes of all the solids of which it is composed (a cylinder and two cones). The silo is made up of a cylinder (with height 10 feet and base radius 5 feet) and two cones (each with height 5 ft and base radius 5 ft). The formulas given at the beginning of the SAT Math section: Volume of a Cone $$V={1}/{3}Ï€r^2h$$ Volume of a Cylinder $$V=Ï€r^2h$$ can be used to determine the total volume of the silo. Since the two cones have identical dimensions, the total volume, in cubic feet, of the silo is given by $$V_{silo}=Ï€(5^2)(10)+(2)({1}/{3})Ï€(5^2)(5)=({4}/{3})(250)Ï€$$ which is approximately equal to 1,047.2 cubic feet. The final answer is D. Question 10 If $x$ is the average (arithmetic mean) of $m$ and $9$, $y$ is the average of $2m$ and $15$, and $z$ is the average of $3m$ and $18$, what is the average of $x$, $y$, and $z$ in terms of $m$? A) $m+6$B) $m+7$C) $2m+14$D) $3m + 21$ ANSWER EXPLANATION: Since the average (arithmetic mean) of two numbers is equal to the sum of the two numbers divided by 2, the equations $x={m+9}/{2}$, $y={2m+15}/{2}$, $z={3m+18}/{2}$are true. The average of $x$, $y$, and $z$ is given by ${x + y + z}/{3}$. Substituting the expressions in m for each variable ($x$, $y$, $z$) gives $$[{m+9}/{2}+{2m+15}/{2}+{3m+18}/{2}]/3$$ This fraction can be simplified to $m + 7$. The final answer is B. Question 11 The function $f(x)=x^3-x^2-x-{11/4}$ is graphed in the $xy$-plane above. If $k$ is a constant such that the equation $f(x)=k$ has three real solutions, which of the following could be the value of $k$? ANSWER EXPLANATION: The equation $f(x) = k$ gives the solutions to the system of equations $$y = f(x) = x^3-x^2-x-{11}/{4}$$ and $$y = k$$ A real solution of a system of two equations corresponds to a point of intersection of the graphs of the two equations in the $xy$-plane. The graph of $y = k$ is a horizontal line that contains the point $(0, k)$ and intersects the graph of the cubic equation three times (since it has three real solutions). Given the graph, the only horizontal line that would intersect the cubic equation three times is the line with the equation $y = −3$, or $f(x) = −3$. Therefore, $k$ is $-3$. The final answer is D. Question 12 $$q={1/2}nv^2$$ The dynamic pressure $q$ generated by a fluid moving with velocity $v$ can be found using the formula above, where $n$ is the constant density of the fluid. An aeronautical engineer users the formula to find the dynamic pressure of a fluid moving with velocity $v$ and the same fluid moving with velocity 1.5$v$. What is the ratio of the dynamic pressure of the faster fluid to the dynamic pressure of the slower fluid? ANSWER EXPLANATION: To solve this problem, you need to set up to equations with variables. Let $q_1$ be the dynamic pressure of the slower fluid moving with velocity $v_1$, and let $q_2$ be the dynamic pressure of the faster fluid moving with velocity $v_2$. Then $$v_2 =1.5v_1$$ Given the equation $q = {1}/{2}nv^2$, substituting the dynamic pressure and velocity of the faster fluid gives $q_2 = {1}/{2}n(v_2)^2$. Since $v_2 =1.5v_1$, the expression $1.5v_1$ can be substituted for $v_2$ in this equation, giving $q_2 = {1}/{2}n(1.5v_1)^2$. By squaring $1.5$, you can rewrite the previous equation as $$q_2 = (2.25)({1}/{2})n(v_1)^2 = (2.25)q_1$$ Therefore, the ratio of the dynamic pressure of the faster fluid is $${q2}/{q1} = {2.25 q_1}/{q_1}= 2.25$$ The final answer is 2.25 or 9/4. Question 13 For a polynomial $p(x)$, the value of $p(3)$ is $-2$. Which of the following must be true about $p(x)$? A) $x-5$ is a factor of $p(x)$.B) $x-2$ is a factor of $p(x)$.C) $x+2$ is a factor of $p(x)$.D) The remainder when $p(x)$ is divided by $x-3$ is $-2$. ANSWER EXPLANATION: If the polynomial $p(x)$ is divided by a polynomial of the form $x+k$ (which accounts for all of the possible answer choices in this question), the result can be written as $${p(x)}/{x+k}=q(x)+{r}/{x+k}$$ where $q(x)$ is a polynomial and $r$ is the remainder. Since $x + k$ is a degree-1 polynomial (meaning it only includes $x^1$ and no higher exponents), the remainder is a real number. Therefore, $p(x)$ can be rewritten as $p(x) = (x + k)q(x) + r$, where $r$ is a real number. The question states that $p(3) = -2$, so it must be true that $$-2 = p(3) = (3 + k)q(3) + r$$ Now we can plug in all the possible answers. If the answer is A, B, or C, $r$ will be $0$, while if the answer is D, $r$ will be $-2$. A. $-2 = p(3) = (3 + (-5))q(3) + 0$$-2=(3-5)q(3)$$-2=(-2)q(3)$ This could be true, but only if $q(3)=1$ B. $-2 = p(3) = (3 + (-2))q(3) + 0$$-2 = (3-2)q(3)$$-2 = (-1)q(3)$ This could be true, but only if $q(3)=2) C. $-2 = p(3) = (3 + 2)q(3) + 0$$-2 = (5)q(3)$ This could be true, but only if $q(3)={-2}/{5}$ D. $-2 = p(3) = (3 + (-3))q(3) + (-2)$$-2 = (3 - 3)q(3) + (-2)$$-2 = (0)q(3) + (-2)$ This will always be true no matter what $q(3)$ is. Of the answer choices, the only one that must be true about $p(x)$ is D, that the remainder when $p(x)$ is divided by $x-3$ is -2. The final answer is D. Want to improve your SAT score by 160 points? We've written a guide about the top 5 strategies you must be using to have a shot at improving your score. Download it for free now: You deserve all the naps after running through those questions. What Do the Hardest SAT Math Questions Have in Common? It’s important to understand what makes these hard questions â€Å"hard.† By doing so, you’ll be able to both understand and solve similar questions when you see them on test day, as well as have a better strategy for identifying and correcting your previous SAT math errors. In this section, we’ll look at what these questions have in common and give examples of each type. Some of the reasons why the hardest math questions are the hardest math questions is because they: #1: Test Several Mathematical Concepts at Once Here, we must deal with imaginary numbers and fractions all at once. Secret to success: Think of what applicable math you could use to solve the problem, do one step at a time, and try each technique until you find one that works! #2: Involve a Lot of Steps Remember: the more steps you need to take, the easier to mess up somewhere along the line! We must solve this problem in steps (doing several averages) to unlock the rest of the answers in a domino effect. This can get confusing, especially if you're stressed or running out of time. Secret to success: Take it slow, take it step by step, and double-check your work so you don't make mistakes! #3: Test Concepts That You Have Limited Familiarity With For example, many students are less familiar with functions than they are with fractions and percentages, so most function questions are considered â€Å"high difficulty† problems. If you don't know your way around functions, this would be a tricky problem. Secret to success: Review math concepts that you don't have as much familiarity with such as functions. We suggest using our great free SAT Math review guides. #4: Are Worded in Unusual or Convoluted Ways It can be difficult to figure out exactly what some questions are asking, much less figure out how to solve them. This is especially true when the question is located at the end of the section, and you are running out of time. Because this question provides so much information without a diagram, it can be difficult to puzzle through in the limited time allowed. Secret to success: Take your time, analyze what is being asked of you, and draw a diagram if it's helpful to you. #5: Use Many Different Variables With so many different variables in play, it is quite easy to get confused. Secret to success: Take your time, analyze what is being asked of you, and consider if plugging in numbers is a good strategy to solve the problem (it wouldn't be for the question above, but would be for many other SAT variable questions). The Take-Aways The SAT is a marathon and the better prepared you are for it, the better you'll feel on test day. Knowing how to handle the hardest questions the test can throw at you will make taking the real SAT seem a lot less daunting. If you felt that these questions were easy, make sure not underestimate the effect of adrenaline and fatigue on your ability to solve problems. As you continue to study, always adhere to the proper timing guidelines and try to take full tests whenever possible. This is the best way to recreate the actual testing environment so that you can prepare for the real deal. If you felt these questions were challenging, be sure to strengthen your math knowledge by checking out our individual math topic guides for the SAT. There, you'll see more detailed explanations of the topics in question as well as more detailed answer breakdowns. What’s Next? Felt that these questions were harder than you were expecting? Take a look at all the topics covered in the SAT math section and then note which sections were particular difficulty for you. Next, take a gander at our individual math guides to help you shore up any of those weak areas. Running out of time on the SAT math section? Our guide will help you beat the clock and maximize your score. Aiming for a perfect score? Check out our guide on how to get a perfect 800 on the SAT math section, written by a perfect-scorer. Want to improve your SAT score by 160 points? Check out our best-in-class online SAT prep classes. We guarantee your money back if you don't improve your SAT score by 160 points or more. Our classes are entirely online, and they're taught by SAT experts. If you liked this article, you'll love our classes. Along with expert-led classes, you'll get personalized homework with thousands of practice problems organized by individual skills so you learn most effectively. We'll also give you a step-by-step, custom program to follow so you'll never be confused about what to study next. Try it risk-free today:

Effect of Color Vision on Foraging and predatoe detection skills in Research Paper

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