List steps please

Question

24. A state’s population as a function of time t (in years) is given by P(t) = 44,178,000 + 78,000t

and its health care costs are given by

C(t) = 47,863,000 + 430,500t + 178t2

a. Create the function representing the state’s per capita health care costs (i.e., the average cost per citizen).

b. Find the average rate of change in the per capita health care cost from year 4 to year 8.

Answer 1

C(t) = 47,863,000 + 430,500t + 178t²

P(t) = 44,178,000 + 78,000t

a.) To find the state's per capita health care costs, we need to divide the cost function by the population function. Let's simplify by using some factoring technique. Here we have:

178t² + 430,500t + 47,863,000 = 2*(89t² + 215,250t + 23,931,500)

78,000t + 44,178,000 = 6000*(13t + 7363)

C(t)/P(t) = 2*(89t² + 215,250t + 23,931,500) / 6000*(13t + 7363)

= 89t² + 215,250t + 23,931,500 / 3000*(13t + 7363)

b.) To find the average rate of change for the per capita health care costs, use the following:

Average ROC = [C(t₂) - C(t₁)] / t₂ - t₁ = [C(8) - C(4)] / 8 - 4 = (51,318,392 - 49,587,848) / 4 = 432,636

The average ROC for the per capita health care costs is about $432,636 per citizen.