According to data from a medical association, the rate of change in the number of hospital outpatient visits, in millions, in a certain country each year from 1980 to the present can be approximated by f’(t)=0.001166t(t-1980)^0.50 , where t is the year.
a. Using The fact that in 1980 there were 256,755,000 outpatient visits, find a formula giving the approximate number of outpatient visits as a function of time.
b. Use the answer to part a to forecast the number of outpatient visits in the year 2015
a. To find the formula for the approximate number of outpatient visits as a function of time, we need to find the antiderivative of f'(t) with respect to t:
f(t) = ∫0.001166t(t-1980)^0.50 dt
f(t) = 0.001166 * (2/3)(t-1980)^(1.50) + C
Given that f(1980) = 256,755,000, we can solve for C:
256,755,000 = 0.001166 * (2/3)(1980-1980)^(1.50) + C
256,755,000 = 0.001166 * 0 + C
C = 256,755,000
Therefore, the formula for the approximate number of outpatient visits as a function of time is:
f(t) = 0.001166 * (2/3)(t-1980)^(1.50) + 256,755,000
b. To forecast the number of outpatient visits in the year 2015, we simply plug in t = 2015 into the formula:
f(2015) = 0.001166 * (2/3)(2015-1980)^(1.50) + 256,755,000
f(2015) = 0.001166 * (2/3)(35)^(1.50) + 256,755,000
f(2015) ≈ 288,975,428
Therefore, the forecasted number of outpatient visits in the year 2015 is approximately 288,975,428.