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 + 178t^2# (How can it be set up?)

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 5 to year 11.
I understand A as C(t)/P(t). Is the answer: 2(89t^2+215,250t+23,931,500)/6,000(13t+7363)?
I am not sure how to do B.

1 Answer
Douglas K.
Jan 25, 2018

Your understanding of A is correct.

The cost per capita is the total cost divided by the total population:

#(C(t))/(P(t)) = (47,863,000 + 430,500t + 178t^2)/(44,178,000 + 78,000t)#

You can perform polynomial long division but the equation gets ugly. I recommend that you give the above as the answer to part A.

The average rate of change from #t_1# to #t_2# is:

#"Average rate of change" = (f(t_2)-f(t_1))/(t_2-t_1)#

In your case, #t_1 = 5" yr"# and #t_2 = 11" yr"#:

#"Average rate of change" = ((C(t_2))/(P(t_2))-(C(t_1))/(P(t_1)))/(t_2-t_1)#

#"Average rate of change" = ((47,863,000 + 430,500(11) + 178(11)^2)/(44,178,000 + 78,000(11))-(47,863,000 + 430,500(5) + 178(5)^2)/(44,178,000 + 78,000(5)))/(11-5)#

#"Average rate of change" ~~ 0.0077#

The above is the answer to part B.

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