Question

The function f(t) = -0.12t^2 + 0.53t + 30.8 models the U.S. populations in millions, ages 65 and older, where t represents years after 1990. The function g(t) = 0.55t^2 + 11.89t + 105.3 models the total yearly cost of Medicare in billions of dollars, where t represents years after 1990. What does the function g/f represent? Find g/t(15).

A. Cost per person in thousands of dollars. $36.67 thousand
B. Cost per person in thousands of dollars. $0.03 thousand
C. Cost per person in thousands of dollars. $8.64 thousand
D. Cost per person in thousands of dollars. $0.16 thousand

Answer 1

The function g(t)/f(t) represents the cost of Medicare per person. Calculating g(15) and f(15) and then dividing g(15) by f(15) gives the cost per person at t=15. The result is approximately $34.67 thousand, which aligns with option A.

Step-by-step explanation:

The function g(t) represents the total yearly cost of Medicare in billions of dollars, and f(t) models the U.S. population ages 65 and older in millions. Therefore, the ratio g(t)/f(t) represents the cost of Medicare per person in the age group 65 and older, in thousands of dollars. To find g/f(15), we substitute t = 15 into both functions and divide the result of g(15) by f(15).

First, calculate f(15):

• f(15) = -0.12(15)^2 + 0.53(15) + 30.8

Next, calculate g(15):

• g(15) = 0.55(15)^2 + 11.89(15) + 105.3

Then, divide g(15) by f(15) to get the cost per person at t=15:

• g/f(15) = g(15) / f(15)

Plugging in the values, we get:

• g(15) = 0.55(15)^2 + 11.89(15) + 105.3 = 0.55(225) + 178.35 + 105.3 = 123.75 + 178.35 + 105.3 = 407.4
• f(15) = -0.12(15)^2 + 0.53(15) + 30.8 = -0.12(225) + 7.95 + 30.8 = -27 + 7.95 + 30.8 = 11.75
• g/f(15) = 407.4 / 11.75 ≈ 34.6745

Therefore, the cost per person is $34.67 thousand, which is closest to option A: Cost per person in thousands of dollars. $36.67 thousand.