Answer Explanations for: ACT June 2012, Form 70C

Transcription

1 Answer Explanations for: ACT June 2012, Form 70C Mathematics 1) C) A mean is a regular average and can be found using the following formula: (average of set) = (sum of items in set)/(number of items in set). Using this formula, you can calculate the average heart rate by adding together the 8 heart rates and dividing by 8: ( )/8 = See averages. 2) H) The area of a rectangle is equal to its length times its width. Using variables, a = lw. Substitute 144 for the area and 4 for the width to get 144 = 4l. Therefore, l = 144/4 = 36. See plane geometry. 3) D) The prices decreased by = 10 dollars. To find the percent decrease, divide the decrease of 10 dollars by the initial price of the coat, 50 dollars: 10/50 =.2 = 20%. If you answered E, you likely divided by the final price rather than the initial price; remember, percent increase or decrease is always calculated with respect to the initial amount, so you must divide by the initial, not the final, amount. You also could have done this problem by calculating 40/50 =.8 = 80%. A new price that is 80% of the original prices indicates a 20% decrease since 100% 20% = 80%. See percentages. 4) J) Because the problem specifies that the number he orders will be in proportion to the results of the survey, you can set up a proportion. When doing so, make sure your units line up on both sides. In other words, this/that = this/that. In this case, (gray shirts)/(total shirts) = (gray shirts)/(total shirts), so 8/50 = x/500. If it is not apparent at this point that x = 80, multiply both sides by 500 to get x = 80. You could have done this problem more quickly by recognizing that the total Chang plans to order is 10 times the total number of customers surveyed, so the total number of gray shirts ordered should be 10 times the number of customers surveyed who preferred gray shirts. See ratios and proportions. 5) C) The perimeter of a rectangle is equal to twice its length plus twice its width, so the perimeter of the garden is = = 40 feet. If you answered A, you forgot to double both dimensions. If you answered E, you found the garden s area. See plane geometry. 6) H) When modeling a linear situation, the y-intercept (b in y = mx + b) is equal to the fixed amount and the slope (m in y = mx + b) is equal to the variable or per something amount. Therefore, H is the correct answer, since the operational cost is the fixed

2 amount and the material cost is the variable amount. Don t let it confuse you that the correct answer is technically written as y = b + mx rather than y = mx + b. Also, note that both values must be positive since they both contribute to the production cost. See equation building. 7) B) In counting problems, you should consider how many events are occurring and how many ways each event can occur. Here you have 2 events (choosing an answer for each of the two questions). It might be wise at this point to write out slots for each event with multiplication symbols in between them: _ _. Then, populate these slots with how many ways each event can occur. Here, the first event has two possible outcomes, and the second event has five possible outcomes, so 2 5 = 10 possible combinations of answers. See counting and probability. 8) J) Distribute the 2 to both terms in the parentheses to get 2x 12 + x = 36 3x 12 = 36 3x = 48 x = 16. If you found this algebra confusing, you also could have gotten this problem right by plugging in the answer choices. 9) D) Guess and check by plugging in the answer choices is likely the fastest way to solve this problem. Begin with the middle answer and try higher or lower values as necessary if your first guess does not work. Soon you will arrive at D, since (4 + 6)/(9 + 6) = 10/15 = 2/3. Alternatively, you could have done this problem by cross multiplying the initial equation to get a = a a = = 6. 10) F) In a geometric sequence, each term is multiplied by the same number, known as the common ratio, to produce the next term. In this sequence, the common ratio is -2, since each term is multiplied by -2 to produce the next term. Therefore, the sixth term is equal to = 192. If you chose K, please note that this answer choice is correct a very small percentage of the time, so even if you do not know how to do a problem, you are better of guessing among the other answer choices. Only choose Cannot be determined on the Mathematics test if you are certain it is correct. See series and sequences. 11) D) If A is moved 4 units to the right, the new x-coordinate is = 0. If it is moved 9 units up, the new y-coordinate is = 0. Therefore, D is the correct answer. See coordinate geometry. 12) F) The two dimensions of the base are equal to l and w in the equation that is provided in the question. Substitute 5 and 12 for l and w and substitute 510 for v to get 5 12 h = h = 510 h = ) E) Because this problem is essentially a recipe problem, it can be solved by setting up a proportion. When doing so, make sure your units line up on both sides. In other words, this/that = this/that. In this case, (square meters)/(amps) = (square meters)/(amps), so x square meters/150 amps = 1 square meter/.75 amps. Multiply both sides by 150 amps

3 to get x = 200 square meters. You may have also been able to recognize the need to divide 150 by.75 without having to set up a proportion, but when in doubt proportions are the way to go. If you answered C, you multiplied 150 by.75 instead of dividing. You also may have been able to solve this problem without doing any math at all. Since each square meter produces less than one amp, more than 150 square meters are needed to produce 150 amps, and there is only one answer choice greater than 150. See ratios and proportions. 14) G) When you purchase 3 cantaloupes at once, the price per cantaloupe (remember, per means divide) is equal to $3.90/3 = $1.30, which is $1.49 $1.30 = $.19 less than the individual price per cantaloupe. 15) E) Multiply like terms together to simplify. Remember that when you multiply the bases, you add the exponents. 4 3 = 12, a a 4 = a 5, and b 2 b 3 = b 5, so the correct answer is E. If you chose C, you may have forgotten that the variable a has an unwritten exponent of 1. If you answered D, you multiplied the exponents instead of adding them. See exponents and radicals. 16) G) Average the y-coordinates of the two endpoints to get the y-coordinate of the midpoint: (3 + 15)/2 = 9. If you answered H, you found the x-coordinate of the midpoint. If you answered K, you forgot to divide by 2. See coordinate geometry. 17) E) On line segment BC, the time (represented on the x-axis) increases, while the distance walked (represented on the y-axis) remains the same, indicating that she is not walking at all during this time period. Therefore, E is the only option that makes sense. 18) G) If lines m and n are not parallel, you cannot conclude that the angles formed the intersection of m and t are equal to those formed by the intersection of n and t, so H, J, and K can be eliminated. F can be eliminated because these angles are supplementary, not equal. G is correct because these angles must be equal since they are vertical angles. See plane geometry. 19) C) Distribute the -3 to both terms in the parentheses, being especially careful because of the negative: 5 6x + 3 = 8 6x. If you answered B, you forgot to distribute the negative to the -1; remember that a negative times a negative is a positive. 20) J) It is impossible to have a triangle that is both right (featuring a 90 angle) and obtuse (featuring an angle greater than 90 ), since these two angles alone would add up to greater than 180, and the sum of the angles of a triangle must always equal 180. See plane geometry. 21) C) On function notation (f(x)) problems, you must substitute whatever is in the parentheses of f(x) for every x in the equation. In this case, substitute -5 for every x in the equation and evaluate. [-3((-5) 2 + 3(-5) + 2)]/(15(-5) + 15) = [-3( )]/(-75 +

4 15) = [-3(12)]/-60 = -36/-60 = 3/5. The most important thing on this problem is being careful not to make computational errors, as there are many opportunities to do so. For instance, if you chose E, you likely calculated (-5) 2 as -25 rather than 25, possibly because you forgot your parentheses around the -5 when entering it into your calculator. See function notation. 22) G) To find the slope, put the equation into y = mx + b (slope-intercept) form, and the slope will be equal to m, the coefficient of the x variable. 2y = -3x + 6 y = (-3/2)x + 6. Therefore, the slope is equal to -3/2. See coordinate geometry. 23) A) The easiest way to do this problem is to use your calculator to get the decimal equivalent of the expression in the question. You should get that it is equal to approximately Try the different answer choices on your calculator to see which one also yields this decimal equivalent, and you will find that the answer is A. This problem can be solved in a more mathematical way by using the property that =. Using this property, you can rewrite the expression in the question as = 2 3 = - See exponents and radicals. 24) F) 5% is equal to the decimal.05, so plug.05 into the equation for r, the interest rate: I =.05Pt. Now divide both sides by.05p to get t = I/(.05t). If you answered G or H, you likely messed up when converting 5% to a decimal. See percentages. 25) A) Translate this confusing language word by word into an equation. Remember to substitute the value of t for t in the parentheses of f(t) as appropriate, which immediately eliminates C, and D, which do not perform this substitution appropriately. Remember as well that is means equals, 100 more than means + 100, and twice means 2 times, and you should be able to arrive at A as your answer. See equation building and function notation. 26) K) Use the Pythagorean Theorem: = x 2 x 2 = 52 x =. See right triangles. 27) C) Because the sum of the angles in a triangle is 180 and a right angle is 90, the sum of the two nonright angles of a right triangle must also be 90. See plane geometry. 28) G) cos = a/h, so cosb = AB/BC. See basic trigonometry. 29) B) Solve compound function notation problems from the inside out. Begin by finding g(1) by substituting 1 for x in g(x). g(1) = = 2. Since g(1) = 2, f(g(1)) = f(2), so find f(2) by substituting 2 for x in f(x). f(2) = = 5. See function notation. 30) K) The power of 3 must be distributed to both terms, the 3 and the x 2. Remember that when raising a power to a power, you multiply the exponents. Therefore, you have 3 3 (x 2 ) 3 = 27x 6. See exponents and radicals.

5 31) B) If BC is 18 inches long, then EC is half of 18, or 9 inches long, since the two right triangles formed by AE are congruent. Knowing that EC = 9, you can use the Pythagorean Theorem to solve for AE x 2 = 15 2 x 2 = 144 x = 12. You also could have avoided having to use the Pythagorean Theorem by recognizing that is a Pythagorean Triple. Either way, if AE = 12, AD = = 16. Even if you did not know how to do this problem, you should have been able to get it correct just by eyeballing the figure, as 16 is the only option that appears to be close to the length of AD. Remember that despite what is stated in the directions of the test, all figures are drawn to scale. If you answered A, you may have forgotten to add 4 to the length of AE. See right triangles. 32) K) Simply crunch these numbers on your calculator to get decimal equivalents. Remember that on many calculators you must enter a mixed number like 3 as (3 + 3/7). Your calculator will show that the first term is equal to about 3.43, the second term is equal to 3.5, and the third term is equal to 3.4, so K is the proper ordering of the numbers. See inequalities and the number line. 33) B) If you let x be equal to the number of large dogs he had shampooed, then the number of small dogs shampooed is equal to 10 x, since a total of 10 dogs were shampooed. Use the prices from the table to set up an equation: 35x + 20(10 x) = x x = x = 60 x = 4. Another good way of doing this problem is through guess and check. Try the different answer choices until you find the one that yields the total price of $260. Start with the middle answer since you will know if you need a higher or lower value, which limits how many you might have to try. See linear systems and equation building. 34) K) Based on the table, each of the first two haircuts cost $55, a total of $110. The third haircut cost.85 $55 = $ ($55 is multiplied by.85 since a discount of 15% implies that the new price is 85% of the original price. Alternately, you could find 15% of $55 and subtract this value from $55.) Therefore the total cost of the three haircuts was $ = $ See percentages. 35) B) Using the price from the table and the fact that 12 small dogs were shampooed per day at that price, you can set up the following equation: 12 = a20 + b. If a $5 price increase decreases the number of small dogs shampooed by 2, then each dollar increase reduces the number of small dogs shampooed by 2/5. Therefore, the slope of this equation (a) is equal to -2/5. In determining this, it is useful to remember that the slope is always the variable or per-something amount in a linear equation. Substitute this slope into the linear equation above to get 12 = (-2/5)20 + b. Now solve for b. 12 = -8 + b b = 20. Therefore, B is the correct answer. See equation building.

6 36) K) If you do this on your calculator, be careful about parentheses. Better yet, do things one step at a time. Begin with the sub-fraction. 1 + ½ = 1.5, so the sub-fraction is equal to 1/1.5 = 2/3. Therefore, the entire denominator is equal to 1 + 2/3 = 5/3. Therefore, the entire fraction can be written as 1/(5/3), which is equal to 3/5. 37) D) The x-intercept of a function is the x-coordinate at which the function intersects the x-axis. Because this point is on the x-axis, the y-coordinate of this point is 0. Plug in 0 for y in the equation and solve for x. 0 = -2x + 8 2x = 8 x = 4. See coordinate geometry. 38) H) Substitute 32 for h and solve for t. 32 = -16t t. The first step in solving a quadratic is always to set the equation equal to zero. 16t 2 48t + 32 = 0. At this point, you have a number of options for solving the equation. You could graph it on your calculator and find the x-intercepts, you could plug it into the quadratic formula, or you could plug in the answer choices for t and solve via guess and check. Here, we will demonstrate how to solve by factoring. Since each term is a multiple of 16, divide both sides by 16 to get t 2-3t + 2 = 0. To factor this quadratic, find two numbers that add to - 3 and multiply to 2. These numbers are -2 and -1, so the factored form of this quadratic is (x 2)(x 1) = 0. Therefore, the solutions are 1 and 2. The correct answer is 1, not 2. Although these values both represent times at which the rocket s height was 32 feet, 1 represents the time at which its height was 32 feet on its way up, and 2 represents the time at which its height was 32 feet on its way down. See quadratics. 39) D) Solve the first inequality to get 3x > 11 x > 11/3. Solve the second inequality to get x < -10/-2 x < 5. (Remember that you must flip the inequality sign when multiplying or dividing both sides by a negative.) If x > 11/3 and x < 5, D is the appropriate number line. See inequalities and the number line. 40) J) Deal with the east-west and the north-south distances separately. Beginning 7 miles east of the headquarters and travelling 5 miles east and 2 miles west puts the cab at = 10 miles east of the headquarters. Beginning 3 miles south of the headquarters and travelling 4 miles north puts the cab at 1 mile north of the headquarters. If the cab is 10 miles east and 1 mile north of the headquarters, its straight line distance from the headquarters can be found using the Pythagorean Theorem = x = x 2 x 10.05, or about 10 miles from the headquarters. You could have also solved this problem by thinking of it in terms of actual coordinate points, adding to the x-value for distance travelled east and subtracting from the x-value for distance travelled west, adding to the y-value for distance travelled north and subtracting from the y-value for distance travelled south, and treating the headquarters as the origin, (0, 0). See coordinate geometry and right triangles. 41) E) Because you are given the adjacent side and trying to find the opposite side, you should set up an equation using tan, since tan involves the opposite and adjacent sides.

7 Tan = o/a, so tan40= x/100. Multiply both sides by 100 to get x = 100tan40. See basic trigonometry. 42) F) Because you are given the opposite side and trying to find the hypotenuse, you should set up an equation using sin, since sin involves the opposite side and the hypotenuse. Sin = o/h, so sin65 = 125/x. Multiply both sides by x to get xsin65 = 125. Divide both sides by sin65 to get x = 125/sin65. See basic trigonometry. 43) C) If the circle is tangent to (meaning it intersects in one single point with) 3 sides of the rectangle, then its diameter is equal to the length of the short side of the rectangle, 6. If its diameter is 6, then its radius is 3, so its area is equal to 3 2 π= 9π. 44) G) Because the point labeled on the circle and the circle s center point share the same x- coordinate, the distance between these points is equal to the difference in their y- coordinates, 2. Therefore, the circle has a radius of 2. Standard form for a circle in the coordinate plane is (x h) 2 + (y k) 2 = r 2, where (h, k) is the center point and r is the radius. Since the circle s center is (2, 1) and its radius is 2, answer choice G is the correct equation. See coordinate geometry. 45) B) Answer choice A can be eliminated quickly; the shaded region is below (less than) the line, so you need a less than or equal to inequality sign. Now you must determine the equation of the line that passes through points a and b. The y-intercept (b in y = mx + b) is a, since that is where the line intercepts the y-axis, so you can eliminate answer choice D. To calculate the slope, you must recognize that you are given two coordinate points, (0, a) and (b, 0). Now you can use the slope formula to find the slope. Slope = rise/run = (change in y)/(change in x) = (y 2 y 1 )/(x 2 x 1 ). In this case, you have (0 a)/ (b 0), so the slope of the line is a/b; therefore, B is the correct answer. See coordinate geometry and inequalities and the number line. 46) F) The diameter of a circle is equal to twice the circle s radius, so the radius of Circle B is equal to (1/2)2x = x. Since the diameter of circle A is also x, the ratio of these two components is 1:1. See plane geometry and ratios and proportions. 47) E) The area of a trapezoid is equal to its height multiplied by the average of its bases. To find the area of this trapezoid, you must first find the length of NM and the height. Begin by drawing a line from point L that is perpendicular to the base NM. Label the point P where this line intersects NM. This line forms the third side of a triangle, triangle LPM. The side ratios of a triangle are x, x, x, where the hypotenuse is equal to x. Since the hypotenuse of this triangle is equal to 10, you can set up the following equation: x = 10. Solve for x to get x = 10/ Therefore, both LP (the height of the trapezoid) and PM are therefore equal to If PM is equal to 7.07, NM = = 25.07, since NP = KL = 18. Now that you have the lengths of the two bases (25.07 and 18) and the height (7.07), you can calculate the area

8 of the trapezoid by multiplying the height by the average of the bases: 7.07(( )/2) 152. An alternative method to finding the lengths of LP and PM if you forgot the side ratios of a triangle would be to use the Pythagorean Theorem, labeling both LP and PM (the two legs) x. If you did not know how to do this problem, but did know how to find the area of a trapezoid, you could have taken a pretty good guess by estimating the lengths of the bottom base and the height and calculating the area based on these estimations. See right triangles and plane geometry. 48) J) There are 8 pauses between the first and last strikes (from 1-2, 2-3, 3-4, 4-5, 5-6, 6-7, 7-8, and 8-9). Each of these 8 pauses lasts 2 second, for a total of 8 2 = 16 seconds. 49) A) Divide both sides of the first equation by 3 to get b = (2/3)a. If b = (2/3)a, then a > b. Multiply both sides of the second equation by 4 to get b = 2c. If b = 2c, then b > c. Therefore, A is the only true inequality. See inequalities and the number line. 50) F) This problem is asking you to find the sum of an arithmetic sequence. Call the amount Malcolm will save next month x. Since he saves an additional dollar each month, his saving for the 12 th month will be x (It is equal to x + 11 rather than x + 12 because he adds his first dollar to x on the second month of savings, his second dollar on the third month, and so on.) Because the amount he is saving each month is increasing at a constant rate, the average amount he saves per month is equal to the average of his first month savings (x) and his last month savings (x +11). Therefore, his average monthly savings is equal to (x + (x + 11))/2 = x Multiply this average monthly savings by 12 to get his total savings over the 12 months: 12(x + 5.5) = 12x Because he has already saved 100 and wants to save a total of 310, he needs to save a total of 210 over the 12 months. Therefore, 12x + 66 = x = 144 x = 12. Alternatively, you could have solved this problem by guess and check, trying different answers as the first month and calculating his savings each month; however, this process would be time consuming. You also could have solved for the sum of the arithmetic series by setting the following formula equal to 210: n(2x + d(n-1))/2 where x is the first term, d is the common difference (1 in this case), and n is the number of terms (12 in this case). See series and sequences. 51) C) A real number is any number that does not contain the imaginary number i. When i is squared (or raised to any even power), it becomes a real number, since i 2 = -1. Therefore, you need to find the answer that will produce only real numbers and i 2 terms (but no i terms) when multiplied by (a + bi). The correct answer is (a bi), since (a + bi)(a bi) = a 2 b 2 i 2 = a 2 b 2 (-1) = a 2 + b 2. This problem is a lot easier if you recognize that (a + bi)(a bi) is the difference of two squares and therefore results in the middle terms that contain the i dropping off. (The difference of two squares states that (x + a)(x a) = x 2 a 2.) This problem would be very time consuming if you had to multiply (a + bi) by each of the answer choices using the FOIL method to determine which one yielded a real solution. See quadratics.

9 52) G) The 100 cm path from A to B along the diameters comprises 4 diameters, so each semicircle has a diameter of 100/4 = 25 cm. Since a semicircle is half a circle, the distance along the four arcs is equal to the circumference of two circles of diameter 25. Since circumference is equal to πd, the circumference of a circle of diameter 25 is 25π, so the distance along the arcs from A to B is equal to 2 25π = 50π. See circles. 53) C) If you cannot picture this graph, simply graph y = a -x on your calculator using any value greater than 1 for a. For instance graph y = 2 -x. Your calculator will show that any such function is decreasing (meaning that it has a negative slope) throughout its domain, so the correct answer is C. See coordinate geometry. 54) J) Inverse variation means that as one variable gets bigger, the other gets smaller. If y varies inversely by x, the equation is y = k/x, where k is a constant. It is also true that y 1 x 1 = y 2 x 2 for any two ordered pairs (x 1, y 1 ) and (x 2, y 2 ), since the product of the x and y variables for any ordered pair will always be equal to k. Here, because the force (f) varies inversely by the square of the distance (d), the equation must feature a d 2 instead of a d. Hence, using the fact that y 1 x 1 = y 2 x 2, you have fd 2 = fd 2. Substitute the distance of 12 for d on one side of the equation and (because you are looking to double the force) 2f for f on the other side: 12 2 f = 2fd 2 144f = 2fd2 72f = fd 2 72 = d 2 d = = = 6. With a more intuitive understanding of inverse variation, you could have done this problem more simply without relying on equations, recognizing that the square of the new distance must be half the square of the old distance (12) in order to double the force. Therefore, the square of the new distance is half of 12 2, which is equal to 72, so the new distance is equal to = = 6. 55) A) The area of a rhombus is equal to base times height, where the height is an altitude, which is by definition perpendicular to the base. Draw the rhombus, and label its sides 4 and its acute angles 60. Then draw an altitude from one of the obtuse angles. This altitude creates a triangle, in which the hypotenuse is 4 (the side of the rhombus) and the altitude forms the larger of the two legs. The side ratios of a triangle are x, x, 2x. Since the 2x is the longest side, it is equal to the hypotenuse, so 2x = 4. Therefore, x = 2. Since the altitude of the rhombus is equal to the longer of the two legs, it is the x side, in this case equal to 2. Since the rhombus has a base of 4 and an altitude of 2, its area is equal to 4 2 = 8. If you did not know the side ratios of a triangle, you could have also used trigonometry to find the length of the altitude. See right triangles and plane geometry. 56) J) The unknown leg of this triangle is 12, since it is a Pythagorean Triple. If you did not recognize this triple, you also could have used the Pythagorean Theorem to find this leg. This cone formed by the rotation described in the question would have a height of 5 and a base radius of 12. Plug these to values into the given equation for h and r respectively and solve. V = (1/3)π = 240π. See right triangles.

10 57) D) As is always the case when subtracting fractions, find a common denominator. In this case, that common denominator is 11 21, since = To get this common denominator, you must multiply both the numerator and the denominator of the left fraction by 11. This gives you 11/ /11 21 = (11 1)/11 21 = 10/ Alternatively, you could have simply done this problem on your calculator, finding the decimal equivalent, and then using your calculator to see which answer choice yields the same decimal. 58) F) Because the range for specified by the inequality is the first quadrant, you do not have to worry about the unit circle and negative values (in other words, you can disregard the inequality if you don t understand its meaning. The easiest way to do this problem is to recognize that since sin /cos = tan (this is an identity you should know), sin = tan cos, so the answer is simply 2/3. If you did not see this trick or did not know the necessary identity, you could still solve the problem. Draw a right triangle, and label one of the nonright angles. Then label the opposite side 2 and the hypotenuse 3, since sin = o/h. Use the Pythagorean Theorem to find the adjacent side: a =. Knowing the adjacent side, you now know that cos = /3 and tan = 2/. Therefore, tan cos = (2/ )( /3) = 2/3. Finally, you could have solved by taking sin - 1 (2/3) on your calculator to find the value of, then calculated tan cos on your calculator using this value. See basic trigonometry and advanced trigonometry. 59) E) Draw an equilateral triangle, and then draw a smaller equilateral triangle inside of it by connecting the midpoints of each of the original triangle s three sides. Your initial triangle is now divided into four smaller equilateral triangles as specified by the problem. The side length of each of these triangles is half the side length of the original triangle, so the perimeters of each of these triangles is also half that of the original triangle. See plane geometry. 60) F) A lot of people will get confused on this problem, thinking that it makes no sense since an absolute value can never be negative. While this is true, it is crucial to note that -x does not necessarily refer to a negative number. Instead, it simply refers to a number with the opposite sign (positive or negative) as x. Therefore, when x itself is negative, -x is positive. Since an absolute value must yield a positive number or zero, x must be negative or 0, so F is the correct answer. See absolute values.