Part Three PRACTICE TESTS

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1 Part Three PRACTICE TESTS

2 HOW TO TAKE THE PRACTICE TESTS Before taking a practice test, find a quiet room where ou can work uninterrupted for one hour. Make sure ou have several No. pencils with erasers. Use the answer grid provided to record our answers. Guidelines for scoring our test appear on the reverse side of the answer grid. Time ourself. Spend no more than one hour on the 50 questions. Once ou start the practice test, don't stop until ou've reached the one-hour time limit. You'll find an answer ke and complete answer eplanations following the test. Be sure to read the eplanations for all questions, even those ou answered correctl. Good luck!

3 HOW TO CALCULATE YOUR SCORE Step : Figure out our raw score. Use the answer ke to count the number of questions ou answered correctl and the number of questions ou answered incorrectl. (Do not count an questions ou left blank.) Multipl the number wrong b 0.5 and subtract the result from the number correct. Round the result to the nearest whole number. This is our raw score. SAT Subject Test: Mathematics Level Practice Test Number Number Raw right wrong score ( 0.5 ) = Step : Find our scaled score. In the Score Conversion Table below, find our raw score (rounded to the nearest whole number) in one of the columns to the left. The score directl to the right of that number will be our scaled score. A note on our practice test scores: Don t take these scores too literall. Practice test conditions cannot precisel mirror real test conditions. Your actual SAT Subject Test: Mathematics Level score will almost certainl var from our practice test scores. However, our scores on the practice tests will give ou a rough idea of our range on the actual eam. Conversion Table Raw Scaled Raw Scaled Raw Scaled Raw Scaled

4 Answer Grid Practice Test # right # wrong Use the answer ke following the test to count up the number of questions ou got right and the number ou got wrong. (Remember not to count omitted questions as wrong.) How to Calculate Your Score on the back of this page will show ou how to find our score.

5 50 Questions ( hour) Directions: For each question, choose the BEST answer from the choices given. If the precise answer is not among the choices, choose the one that best approimates the answer. Then fill in the corresponding oval on the answer sheet. Notes: Practice Test () To answer some of these questions, ou will need a calculator. You must use at least a scientific calculator, but programmable and graphing calculators are also allowed. () Make sure our calculator is in the correct mode (degree or radian) for the question being asked. () Figures in this test are drawn as accuratel as possible UNLESS it is stated in a specific question that the figure is not drawn to scale. All figures are assumed to lie in a plane unless otherwise specified. (4) The domain of an function f is assumed to be the set of all real numbers for which f() is a real number, unless otherwise indicated. Reference Information: Use the following formulas as needed. Right circular cone: If r = radius and h = height, then Volume = πr h; and if c = circumference of the base and l = slant height, then Lateral Area = c l. Sphere: If r = radius, then Volume = 4 πr and Surface Area = 4πr. Pramid: If B = area of the base and h = height, then Volume = Bh. 5

6 6 Practice Tests. If = 7 5, what is the value of? DO YOUR FIGURING HERE. (A). (B).6 (C) 5.6 (D) 4.0 (E) 6,04.6. If a b c = ab c, which of the following equals 5? (A) 4 (B) 5 5 (C) 6 4 (D) 8 4 (E) 0 4. If f() = e and g() =, then g(ƒ()) = (A).7 (B).7 (C) 4. (D) 5.4 (E) If + = 5, what is the value of? (A) (B) (C) (D) (E) 4 GO ON TO THE NEXT PAGE

7 Practice Test 7 5. In Figure, if cos θ = 0.75, tan θ = DO YOUR FIGURING HERE. (A) 0.60 (B) 0.67 (C) 0.75 (D) 0.88 (E). 6. Which of the following is an equation of the line that has a -intercept of 6 and an -intercept of? (A) = 6 (B) = 6 (C) + = 6 (D) 6 + = (E) 6 = θ Figure 7. For all 0, + + = (A) (B) (C) (D) (E) GO ON TO THE NEXT PAGE

8 8 Practice Tests 8. In a class of 0 bos and 5 girls, the average score on a biolog test is 90. If the average score for the girls is, what is the average score for the bos in terms of? DO YOUR FIGURING HERE. (A) 00 (B) 5 (C) 50 (D) 50 (E) Which of the following graphs is smmetric about the origin? (A) (B) (C) (D) (E) GO ON TO THE NEXT PAGE

9 Practice Test 9 0. George is going on vacation and wishes to take along books to read. If he has 5 different books to choose from, how man different combinations of books can he bring? DO YOUR FIGURING HERE. (A) (B) 5 (C) 0 (D) 5 (E) 0. If =, = (A) or 6 (B) or 6 (C) onl (D) 6 onl (E) There is no solution.. The lines with the equations = m + 4 and = m + will intersect to the right of the -ais if and onl if (A) m = m (B) m < m (C) m > m (D) m + m = 0 (E) m m. If the probabilit that it will rain sometime on Monda is and the independent probabilit that it will rain sometime on Tuesda is, what is the probabilit that it will rain on both das? (A) 6 (B) 5 (C) (D) 5 (E) 5 6 GO ON TO THE NEXT PAGE

10 0 Practice Tests 4. If sin A =, then sin Acos A = DO YOUR FIGURING HERE. (A) (B) (C) (D) (E) 4 5. What values for would make undefined? + (A) onl (B) onl (C) All real numbers greater than (D) All real numbers less than (E) All real numbers less than or equal to 6. If, the maimum value of f() = is (A) (B) (C) 0 (D) (E) 7. If f() = 4, and f () is the inverse of f(), then ƒ( ) ƒ ( ) = (A) (B) (C) 4 (D) 0 (E) 8. Which of the following polnomials, when divided b + 4, equals + 5 with remainder? (A) 6 + (B) (C) (D) (E) GO ON TO THE NEXT PAGE

11 Practice Test 9. Let be defined to be the floor of, where is the DO YOUR FIGURING HERE. greatest integer that is less than or equal to, and let be the ceiling of, where is the least integer that is greater than or equal to. If f() = + and is not an integer, then f() is also equal to (A) (B) (C) (D) (E) If log + log = 7, then = (A). (B).40 (C). (D) 8.5 (E).6. If f ( )=, what is the domain of f()? (A) All real numbers ecept for 0 (B) All real numbers greater than or equal to (C) All real numbers less than or equal to (D) All real numbers greater than or equal to but less than or equal to (E) All real numbers less than or equal to. How man was can identical red chairs and 4 identical blue chairs be arranged in one row? (A) 6 (B) 5 (C) (D) 4 (E) 0 GO ON TO THE NEXT PAGE

12 Practice Tests. If a + b > 0 and c + d > 0, which of the following must be true? DO YOUR FIGURING HERE. (A) a + b + c > 0 (B) ac + bd > 0 (C) a + b > 0 (D) d(a + b) > 0 (E) a + b > c + d 4. If > 0, a = cos θ, and b = sin θ, then a + b = (A) (B) (C) (D) (cos θ + sin θ) (E) cos θ sin θ 5. If 0 < < 90 and 5 sin = 7 sin, what is the value of sin? (A).00 (B) 0.7 (C) 0.40 (D) 0.8 (E) i and + i are roots to which of the following quadratic equations? (A) = 0 (B) 6 + = 0 P(,) (C) 6 = 0 (D) = 0 (E) = 0 O In Figure, if point P is located on the unit circle, then + = (A) 0.7 (B) 0.50 (C) 0.78 (D) 0.87 (E).7 Figure GO ON TO THE NEXT PAGE

13 Practice Test 8. If 0 t, which of the following graphs is the graph of versus where and are related b the parametric equations = t and = t? DO YOUR FIGURING HERE. (A) (B) (C) (D) (E) GO ON TO THE NEXT PAGE

14 4 Practice Tests 9. In Figure, if isosceles right triangle ABC and square ACDE share side AC, what is the degree measure of angle EBC? DO YOUR FIGURING HERE. B (A) 7 (B) 0 (C) 60 (D) 6 (E) In Figure 4, which of the following denotes the correct vector arithmetic? A C (A) + = z (B) + z = (C) + z = E D (D) (E) z = z = Figure. The horizontal distance, in feet, of a projectile that is fired with an initial velocit v, in feet per second, at an angle θ with the horizontal, is given b H(v, θ) = v sin( θ ) If a football is kicked at an angle of 50 degrees with the horizontal and an initial velocit of 0 feet per second, what is the horizontal distance, in feet, from the point where the football is kicked to the point where the football first hits the ground? (A) 8 (B) 0 (C) (D) 6 (E) 9 z Figure 4. If a right circular cone has a lateral surface area of 6π and a slant height of 6, what is the radius of the base? (A) 0.50 (B) 0.75 (C).00 (D).5 (E).50 GO ON TO THE NEXT PAGE

15 Practice Test 5. If two fair dice are tossed, what is the probabilit that the two numbers that turn up are consecutive integers? DO YOUR FIGURING HERE. (A) 0.4 (B) 0.7 (C) 0.8 (D) 0. (E) Which of the following is an equation of the ellipse centered at (,) with a minor ais of 4 parallel to the -ais and a major ais of 6 parallel to the -ais? (A) (B) (C) (D) (E) ( ) ( ) ( + ) ( ) ( ) ( + ) ( + ) ( + ) ( ) ( + ) = = = = = 5. If f() 0 and g() 0 for all real, which of the following statements must be true? I. f() + g() 0 II. f() g() 0 III. f()g() 0 (A) (B) (C) (D) (E) I onl II onl III onl I and II I and III GO ON TO THE NEXT PAGE

16 6 Practice Tests 6. Where defined, sinθ cscθ (A) sin θ (B) csc θ (C) sin θ (D) csc θ (E) cos θ DO YOUR FIGURING HERE. 5 k 7. ( ) k k= 0 (A) 0 (B) 6 (C) 0 (D) 6 (E) 0 8. If f() =, which of the following must be true? (A) (B) (C) (D) (E) f( ) = f() f( ) = f( ) f( ) = f() f() = f() f() > f( ) lim + 4 = (A).5 (B) 0.80 (C) 0.80 (D).5 (E) The limit does not eist. 40. If Figure 5 shows the graph of f(), what is the value of f( f ())? 4 4 (A) 4 (B) (C) 0 (D) (E) Figure 5 GO ON TO THE NEXT PAGE

17 Practice Test 7 4. If all the terms of a geometric series are positive, the first term of the series is, and the third term is 8, how man digits are there in the 40th term? DO YOUR FIGURING HERE. (A) 0 (B) (C) (D) (E) 4 4. In Figure 6, what is the degree measure, to the nearest integer, of angle ABO? z A(0,0,) (A) 50 (B) 48 (C) 45 (D) 4 (E) 40 O B(,,0) 4. If log ( 6) = log 4 ( 4), which of the following could be the value of? Figure 6 (A) (B) (C) 6 (D) 0 (E) If a sphere of radius is inscribed in a cube such that it is tangent to all si faces of the cube, the volume contained outside the sphere and inside the cube is (A) 97 (B) 0 (C) 09 (D) 5 (E) 45. If f() = sin (arctan ), g() = tan (arcsin ), and 0 < π, then f g π 0 = (A) 0.4 (B) 0.54 (C) (D) (E) GO ON TO THE NEXT PAGE

18 8 Practice Tests 46. If f ( ) =, what is the smallest integer such that ( + )! f() < ? DO YOUR FIGURING HERE. (A) 7 (B) 8 (C) 9 (D) 0 (E) 47. In Figure 7, point O has coordinates (0,0), point P lies on the graph of = 6, and point B has coordinates (,0). If OP = BP, the area of triangle OPB is (A).7 (B).0 (C).5 (D) 4.7 (E) 5. = 6 P O B 48. If cos = sin, and is in radians, which of the following is a possible value of? Figure 7 (A) 0.9 (B) 0.5 (C).05 (D).60 (E) In Figure 8, if a wooden right circular clinder with radius meters and height 6 meters has a clindrical hole of diameter meters drilled through the center as shown, what is the entire surface area (including the top and bottom faces), in square meters, of the resulting figure? (A) (B) (C) (D) (E) 8π 40π 4π 44π 46π Figure What is the greatest possible number of points of intersection between a parabola and a circle? (A) (B) (C) 4 (D) 6 (E) 8 STOP! If ou finish before time is up, ou ma check our work.

19 Turn the page for answers and eplanations to Practice Test.

20 Answer Ke Practice Test. C. E. B 4. C 5. D 6. B 7. D 8. B 9. E 0. C. C. B. A 4. C 5. E 6. B 7. E 8. E 9. D 0. C. B. B. C 4. B 5. C 6. B 7. E 8. C 9. A 0. C. A. C. C 4. B 5. E 6. C 7. B 8. C 9. B 40. C 4. D 4. E 4. D 44. B 45. A 46. B 47. E 48. B 49. C 50. C 0

21 Practice Test Answers and Eplanations ANSWERS AND EXPLANATIONS. C Use our calculator. First find that 7 5 = 6,807. Then find that the cube root of 6,807 is about E Go down the answer choices and tr multipling the first two numbers and dividing b the third until ou find the choice that ields 5. The answer is (E) because 0 = B Perform the inside function first: f ( ) = e f( ) = e 789. Then perform the outside function on the result: g ( ) = g(. 7 89) = D First use our calculator to solve for θ : cos θ = 0.75 θ = arccos (0.75) 4.4 Then find the tangent of the result: 6. B tan (4.4 ) 0.88 A -intercept of 6 means that one point on the line is (0,6). Plug = 0 and = 6 into the answer choices, and ou ll find that onl (B) and (C) work. Now test those two choices with the -intercept (,0). Of (B) and (C), onl (B) works this time. 7. D To add fractions, ou need a common denominator. Here the LCD is 6 : = + + = C Manipulate the equation to get + = 5 + = 5 = = = on one side: 8. B Let represent the average score for the bos. Fifteen girls average, so their 5 scores add up to 5. Ten bos average, so their 0 scores add up to 0. The 5 students average 90, so their scores add up to 5 90 =,50. Now ou can set up an equation and solve for. The girls total and the bos total add up to the grand total of,50: 5+ 0 =, 50 0 =, 50 5, 50 5 = = 5 0

22 Practice Tests 9. E To be smmetric about the origin means that, for an point A on the graph, there is another point B on the graph such that the origin is the midpoint of AB. So ou should be able to start at an point on the graph, draw a straight line segment to the origin, continue straight the same distance beond the origin, and be at another point on the graph. Thus, for eample, ou can see that (A) does not work: But (E) does work:. C One of the steps in solving this equation is to square both sides. That step can result in etraneous solutions. Look: = = + ( ) = ( + ) = = 0 ( + )( + 6) = 0 = or 6 But if ou plug those solutions back into the original equation, ou ll find that = 6 doesn t work. The onl solution is =. Alternativel, ou could Backsolve using the numbers provided in the answer choices. 0. C George has 5 choices for the first book and then 4 choices for the second. That s 0 permutations for taking books out of 5. But this is a combinations question. Order doesn t matter. Whether he reads A or B first, it s the same combination of books, so ou have to divide the 0 permutations b. If ou like formulas, here s how to do this one: n C C r 5 n! = r! n r! ( ) 5! = = 0!!

23 Practice Test Answers and Eplanations. B You can think this one through conceptuall: The first equation intercepts the -ais at (0, 4), and the second equation intercepts the -ais at (0, ). That the intersect to the right of the -ais means that the -coordinate of the point of intersection is positive: Since is positive, so is : m m = m m > 0 m > m l. A l O The probabilit of independent events occurring is the product of the separate probabilities: = C l l O You can use our calculator. Arcsin = 0, so A = 0 and A = 5. Now use our calculator to find that sin5 cos5 =. But the solution s even quicker if ou remember the double-angle sine formula: sin A = sin A cos A Therefore, In both of the cases shown, l has the greater slope, so m > m. Alternativel, ou could do this one algebraicall. The point of intersection is the point at which m + 4 = m + : m+ 4= m+ m m= 4 m ( m ) = m m = 5. E = = = sinacos A sin A 05. The epression will be undefined when the denominator is zero or when the epression under the radical is negative. The denominator is zero when + = 0 + = 0 =

24 4 Practice Tests The epression under the radical is negative when + < 0 < So the epression is undefined for all. 6. B You can think this one through conceptuall. The epression will be at its maimum when the that s subtracted from the is as small as it can be. The square of a real number can t be an smaller than 0 and = 0 is within the specified domain so 7. E maimum = 0 = First find the inverse of f(): f( ) = 4 = 4 + 4= f Now find f( ) and f ( ): f ( ) = 4 4= = 4 ( ) = 4 f( ) = 4( ) = ( ) = f ( ) = 4 4 f = ( ) ( ) = = 4 4 And now, to get f ( )f ( ), multipl: f ( ) f ( )= () = 8. E To find the original polnomial, multipl + 4 b + 5 and then add the remainder to the result: 9. D ( + 4)( + 5 ) + = = When is not an integer, the floor and ceiling are apart. In other words, = + With this ou can epress the definition of the function: 0. C. B log f ( )= + = + + = + + log = 7 log = 7 ( ) log = 5..5 =. To be in the domain of this function, must not be anthing that makes the epression under the radical negative or that makes the denominator zero. The epression under the radical is, and it must be nonnegative: 0 The denominator s simpl, which then cannot be zero. It s alread been established, however, that must be greater than or equal to, so that s the domain.

25 Practice Test Answers and Eplanations 5. B There is a formula that applies to this situation. The number of distinct permutations of n things, a of which are indistinguishable, b of which are indistinguishable, etc., is n! ab!! Here there are 6 chairs, of which are indistinguishable and 4 of which are indistinguishable, so the number of permutations is. C 6! ! 4! = = = 4 The best wa to go about this one is to check out each answer choice, tring to think of a case where that choice is not true. The correct answer is the one that has no countereample. That a + b > 0 and c + d > 0 would impl, for eample, that the total sum a + b + c + d would also be positive, but that s not the same as saing (A), a + b + c > 0. If a =, b =, c = 4, and d = 5, then (A) is not true. Nor is (B). (C) is true for this set of numbers and for an possible set of numbers because a + b will be greater than zero as long as a and b are not both zero. 4. B a + b = ( cos θ) + ( sin θ) = (cos θ + sin ) = = It s given that > 0, so =. 5. C What ou have here is a quadratic equation in which the unknown is sin. To make things simpler, replace sin with and solve for : 5sin = 7sin 5 = = = ± 0 7 = ± = or 0 0 = 040. or It s given that is a positive acute angle, so 0 < sin <, and onl 0.40 fits. 6. B If the solutions to a + b + c = 0 are ± i, then b b 4ac = and = i a a In all the answer choices, a =, so ou can sa more simpl b = b = 6 b 4c = i b 4c = 4i b 4c = 6 6 4c = 6 4c = 5 c = So the answer is the equation with a =, b = 6, and c = : 6 + = 0. Another wa to approach this problem is to generate the epression from the roots. If the solutions are ± i, then the factored epression is [ ( + i)][ ( i)]. Multipling this out gives ( i) ( + i) + ( + i)( i) = + i i + 9 6i+ 6i 4i Combining like terms gives ou i = 6 +

26 6 Practice Tests 7. E Draw in a right triangle: O 60 P(, ) That s a triangle. The hpotenuse is the radius, so it s. That means that the short leg is and the long leg is : O 60 P(, ) 8. C Combine the equations so as to lose t and get in terms of : = = t = t t = ( ) = 4 So ou might be tempted b (E), which looks like the graph of = 4. But the stem sas that 0 t, so the onl possible values of the correct graph is (C). 9. A = t are 0, so Mark up the figure. Call the sides of the square and the legs of the triangle each. What ou re looking for is the measure of the angle marked in right triangle BDE : A B C So, + = E The leg opposite is, and the leg adjacent to is, so D opposite tan = = adjacent = arctan 7

27 Practice Test Answers and Eplanations 7 0. C The figure shows the head of touching the tail of z, so those are the two vectors being added. The result is because it then connects the tail of to the head of z. Therefore,. A + z = Just plug θ = 50 and v = 0 into the formula and crank out the answer:. C H = v sin( θ ) 0 ( 50 ) ( ) sin = 900sin00 = 8 A formula for the lateral area of a cone is given in the directions : Lateral area = cl Here the lateral area is 6π and l = 6, so ou can solve for c : 6π = c() 6 π = c c = π Now ou can use the base circumference c = π to find the base radius: Circumference = πr π = πr r =. C The total number of possible outcomes is 6 6 = 6. Of those outcomes, the following are consecutive integers: and and and and and 4 4 and 4 and 5 5 and 4 5 and 6 6 and 5 That s 0 favorable outcomes: 4. B Favorableoutcomes Probabilit = Totalpossibleoutcomes 0 = The equation of an ellipse centered at the point (p, q) and with aes a and b is ( p) ( q) + a b = Here p =, q =, a =, and b =, so the equation is ( + ) ( ) + which is the same as choice (B). =

28 8 Practice Tests 5. E Don t let the functions smbolism confuse ou. Just think of these as two quantities f() and g() that are both nonnegative. Statement I sas their sum is nonnegative. That s true add an two nonnegatives and ou ll get a nonnegative sum. Statement II sas the difference f() g() is nonnegative. Well, that s true onl if f() g(). But there s no reason that g() couldn t be greater than f(). Statement III sas their product is nonnegative. That s true multipl an two nonnegatives and ou ll get a nonnegative product. Statements I and III are true. 6. C Epress cosecant as over sine: sinθ = sinθ cscθ sinθ sinθ = sinθ sinθ sinθ = sinθ sinθ sinθ ( sin θ)(sin θ) = sinθ = (sinθ )(sin θ) sinθ = sinθ 7. B Just plug in the si possible values for k, compute the results, and add them up: 8. C 0 k = 0 ( ) 0 ( ) = 0 k = ( ) () = k = ( ) ( ) = k = ( ) ( ) = 6 4 k = 4 ( ) 4 ( ) = 8 k = 5 ( ) 5 5 ( )= = 6 The function is cubing. Think about each answer choice. (A): The cube of minus equals the cube of? No. (B): The cube of minus equals the opposite of the cube of minus? No. (C): The cube of minus equals the opposite of the cube of? That sounds plausible. In fact, (C) is true. It doesn t matter whether ou take the opposite of a number first and then cube it, or cube the number first and then take its opposite ou ll get the same result both was.

29 Practice Test Answers and Eplanations 9 9. B The first step in finding a limit is generall to factor: = ( )( ) ( + 4)( ) If ever actuall gets to, the epression becomes undefined zero over zero. But if ou cancel out the ( ) from the top and bottom: ( 5)( ) = 5 ( + 4)( ) + 4 ou can plug in = and find the limit: 40. C 5 lim + = = 4 = Don t tr to figure out an equation to fit this weird graph. Just read the values right off the graph. First, find f(). Go to + on the -ais and see what is there. It s. Now find f ( ). Go to on the -ais and see what is there. It s 0: 4. D The first term is and the third term is, so the 40th term is 40. Use our calculator and ou ll get an answer in scientific notation something like this: That is,.0995 E That s followed b digits, for a total of digits. 4. E The length of OA is, and the length of OB is + =. OA over OB is the tangent of the angle ou re looking for: OA tan = = OB = arctan 40 f( ) = f(()) f = f( ) = 0

30 40 Practice Tests 4. D Put everthing in terms of log : log ( 6) = log ( 4) 4 log ( 4) log ( 6) = log 4 log ( 4) log( 6) = log ( 6) = log ( 4) log ( 6) = log ( 4) ( 6) = = = 0 ( )( 0) = 0 = or 0 Of those two apparent solutions, one is impossible. The log of a negative number is undefined, so = is an etraneous solution: You can t take the log of ( 6). The onl solution is A You could use our calculator and do this one step b step. Set our calculator to radian mode. First perform the inside function: g π π 0 = tan arcsin 0 tan( 0.96) Then perform the outside function on the result: f ( 0. 09) = sin(arctan( 0. 09)) sin( 0. 96) 0. 4 Far quicker and simpler would be to realize that if ou take the sin of the arctan of the tan of the arcsin, ou ll end up back where ou started. The answer to this question is just the decimal approimation of the fraction π B The cube is 6 6 6, so its volume is 6. The sphere has radius, so Volume of sphere = 4 πr The difference is 6 6π 0. 4 = π( ) = 6π 46. B First convert into a fraction: = =, 000, , 000 You re looking for the smallest integer value of that will make ( +)! less than. 00, 000 In other words, ou re looking for the smallest integer that will make ( + )! greater than 00,000. Use our calculator and tr a few possibilities = 40,0. Not big enough. But multipl that b 9 and ou re up to 6,880. So 9! > 00,000, + = 9, and therefore = 8.

31 Practice Test Answers and Eplanations E To find the area of the triangle, ou want the base and the height. The base is the length OB, which is simpl the -coordinate of point B:. The height is the -coordinate of point P, which is equal to 6. The triangle is isosceles, so the altitude from P to base OB divides the base in half and the -coordinate for point P is equation = 6 to find the height: height = 6 ( ) = 6 = So if the base is and the height is : Area= (base)(height) = ( ) =. Plug = into the ( ) C The entire surface area ou re looking for consists of the lateral areas of the outside clinder and the inside clinder, plus the areas of the larger top and bottom circles, minus the areas of the smaller top and bottom circles. The lateral area of the outside clinder is πrh = π()(6) = 4π. The lateral area of the inside clinder is πrh = π()(6) = π. The areas of the larger top and bottom circles are each πr = π( ) = 4π. And the areas of the smaller top and bottom circles are each πr = π( ) = π. The total surface area, then, is 50. C 4π + π + (4π) (π) = 4π Visualize the situation and/or make a few sketches. Tr to imagine as man points of intersection as possible. Here s a wa to get four: 48. B Use the relationship cos = sin to get everthing in terms of sine. And be sure our calculator is in radian mode. cos= sin sin = sin sin + sin = 0 ( sin ) (sin + ) = 0 sin = or = arcsin or arcsin( ) 05. or 57. There s no wa to get more. Of those solutions, onl 0.5 is listed in the answer choices.

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