SAT Math Section: 3 and 4 #2

(Section 3) Question #6: h = 3a + 28.6 A pediatrician uses the model above to estimate the height h of a boy, in inches, in terms of the boy's age a, in years, between the ages of 2 and 5. Based on the model, what is the estimated increase, in inches, of a boy's height each year? A) 3 B) 5.7 C) 9.5 D) 14.3

Answer: A) 3 Explanation: Choice A is correct. In the equation h = 3a + 28.6, if a, the age of the boy, increases by 1 then h becomes h = 3(a+1) + 28.6 =3a + 3 + 28.6 = (3a + 28.6) + 3. Therefore, the model estimates that the boy's height increases by 3 inches each year. Alternatively: The height, h, is a linear function of the age, a, of the boy. The coefficient 3 can be interpreted as the rate of change of the function; in this case, the rate of change can be described as a change of 3 inches in height for every additional year in age. Choices B, C, and D are incorrect and are likely the result of dividing 28.6 is the estimated height, in inches, of a newborn boy. However, dividing 28.6 by 5,3, or 2 has no meaning in the context of this question.

(Section 3) Question #4: Kathy is a repair technician for a phone company. Each week, she receives a batch of phones that need repairs. The number of phones that she has left to fix at the end of each day can be estimated with the equation P d = 108 − 23 , where P is the number of phones left and d is the number of days she has worked that week. What is the meaning of the value 108 in this equation? A) Kathy will complete the repairs within 108 days. B) Kathy starts each week with 108 phones to fix. C) Kathy repairs phones at a rate of 108 per hour. D) Kathy repairs phones at a rate of 108 per day.

Answer: B) Kathy starts each week with 108 phones to fix. Explanation: Choice B is correct. The value 108 in the equation is the value of P in P=108-23d when d=0. When d=0, Kathy has worked 0 days that week. In other words, 108 is the number of phones left before Kathy has started work for the week. Therefore, the meaning of the value 108 in the equation is that Kathy starts each week with 108 phones to fix. Choice A is incorrect because Kathy will complete the repairs when P=0. Since P= 108 - 23d, this will occur when 0= 108 - 23d or when d= 108/23, not when d= 108. Therefore, the value 108 in the equation does not represent the number of days it will take Kathy to complete the repairs. Choices C and D are incorrect because the number 23 in P= 108 - 23d indicates that the number of phones left will decrease by 23 for each increase in the value of d by 1; in other words, Kathy is repairing phones at a rate of 23 per day, not 108 per hour (choice C) or 108 per day (choice D).

(Section 3) Question #14: 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.

Choice A is correct. 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 as 2^3x/2^y. Since the numerator and denominator 2^3x/2^y have a common base, this expression can be rewritten as 2^3x - y. It is given that 3x - y = 12, so one can substitute 12 for the exponent, 3x - y, given that the expression 8^x/2^y is equal to 2^12. Choice B is incorrect. The expression 8^x/2^y can be rewritten as 2^3x/26Y, OR 2^3X-Y. If the value of 2^3X-Y is 4^4, which can be rewritten as 2^3x/2^y, or 2^3x-y = 2^8, which results in 3x - y = 8, not 12. Choice C is incorrect. If the value of 8^x/2^y is 8^2, then 2^3x-y = 8^2, which results in 3x - y = 6, not 12. Choice D is incorrect because the value of 8^x/2^y can be determined.

(Section 3) Question #12: A line in the xy-plane passes through the origin and has a slope of 1/7 . Which of the following points lies on the line? A) (0, 7) B) (1, 7) C) (7, 7) D) (14, 2)

Choice D is correct. Explanation: In the xy-plane, all lines that pass through the origin are of the form y = mx, where m is the slope of the line. Therefore, the equation of this line y= 1/7 x, or x = 7y. A point with coordinates (a, b) will lie on the line if and only if a =7b. Of the given choices, only choice D, (14,2) satisfies this condition: 14 = 7(2). Choice A is incorrect because the line determined by the origin (0,0) and (0,7) is the vertical line with equation x =0; that is, the y-axis. The slope of the y-axis is undefined, not 1/7. Therefore, the point (0,7) does not lie that passes the origin and has slope 1/7. Choices B and C are incorrect because neither of the ordered pairs has a y-coordinate that is 1/7 the value of the corresponding x-coordinate.

(Section 3) Question #15:

Choice D is correct. Explanation: One can find the possible values of a and b in (ax + 2) (bx + 7) by using the given equation a + b = 8 and finding another equation that relates the variables a and b. Since (ax + 2)(bx + 7) = 15x^2 + cx + 14, one can expand the left side of the equation to obtain abx^2 + 7ax + 2bx + 14 = 15x^2 + cx + 14. Since ab is the coefficient of x^2 on the left side of the equation and 15 is the coeffiecent of x^2 on the right side of the equation, it must be true that ab = 15. Since a + b = 8, it follows that b = 8- a. Thus, ab = 15 can be rewritten as a 8 - a) = 15, which in turn can be rewritten as a^2 - 8a + 15 = 0. Factoring gives (a -3)(a - 5) = 0. Thus, either a = 3 and b = 5, or a = 5 and b = 3. If a = 3 and b = 5, then (ax + 2)(bx + 7) = (3x + 2)(5x + 7) = 15x^2 + 31x + 14. Thus, one of the possible values of c is 31. If a = 5 and b = 3, then (ax + 2)(bx + 7) = (5x + 2)(3x + 7) = 15x^2 + 41x + 14. Thus, another possible value for c is 41. Therefore, the two possible values for c are 31 and 41.

(Section 3) Question #11: b= 2.35 + 0.25x c= 1.75 + 0.40x In the equations above, b and c represent the price per pound, in dollars, of beef and chicken, respectively, x weeks after July 1 during last summer. What was the price per pound of beef when it was equal to the price per pound of chicken? A) $2.60 B) $2.85 C) $2.95 D) $3.35

Choice D is correct. Explanation: To determine the price per pound of beef when it was equal to the price per pound of chicken, determine the value of [ (the number of weeks after July 1) when the two prices were equal. The prices were equal when b = c; that is, when 2.35 + 0,25x= 1.75 + 0.40x. This last equation is equivalent to 0.60 = 0.15x, and so x = 0.69/0.15 = 4. Then to determine b, the price per pound of beef, substitute 4 for x in b = 2.35 + 0.25x, which gives b = 2.35 + 0.25(4) = 3.35 dollars per pound. Choice A is incorrect. It results from substituting the value 1, not 4, for x in b = 2.35 + 0.25x. Choice B is incorrect. It results from substituting the value 2, not 4, for x in b 2.35 + 0.25x. Choice C is incorrect. It results from substituting the value3, not 4, for x in c = 1.75 + 0.40x.

(Section 3) Question #17: *(Look at paper card for image of problem)

The correct answer is 1600. It is given that ∠$(% and ∠&'% have the same measure. Since ∠$%( and ∠&%' are vertical angles, they have the same measure. Therefore, triangle ($% is similar to triangle '&% because the triangles have two pairs of congruent corresponding angles (angle-angle criterion for similarity of triangles). Since the triangles are similar, the corresponding sides are in the same proprtion; thus CD/x = BD/EB. Substituting the given values of 800 for CD, 700 for BD, and 1400 for EB in CD/x = BD/EB gives 800/x = 700/1400. Therefore, x = (800)(1400)/700 = 1600.

(Section 3) Question #18:

The correct answer is 7. Subtracting the left and right sides of x + y = -9 from the corresponding sides of x + 2y = -25 gives (x + 2y) - (x + y = -25 -(-9)), which is quivalent to y = -16. Substituting -16 for y in x + y = -9 gives x + (-16) = -9, which is equivalent to x = -9 - (-16) = 7