Based on the current eligibility requirement, a study conducted in 2004 showed that federal spending on entitlement programs, particularly Medicare, would grow enormously in the future. The study predicted that spending on Medicare, as a percentage of the gross domestic product (GDP), will be P(t) = 0.27t^2 + 1.4t + 2.1 (0 <= t <= 5) percent in year t, where t is measured in decades, and t = 0 corresponds to 2000. (Round your answers to two decimal places.)

a. How fast will the spending on Medicare, as a percent of the GDP, be growing in 2030?
b. What will the predicted spending on Medicare be in 2030?

A) 2.9% of GDP

B) 8.73% of GDP

Step-by-step explanation:

A) To find the rate of spending, let's find the derivative of the function

P(t) = 0.27t² + 1.4t + 2.1 (0 <= t <= 5)

So,

dP/dt = 0.54t + 1.4 which is the annual growth rate.

Since t is measured in decades, thus in 2030 ,it will have been 3 decades, so we have t = 3

So, at t=3; dp/dt = 0.5(3) + 1.4 = 1.5 + 1.4 = 2.9 which means spending in 2030 will be growing at rate of 2.9%

B) The predicated spending is;

P(t) = 0.27t² + 1.4t + 2.1

Thus, in 2030, will be 3 decades so, t= 3

Thus;

P(3) = 0.27(3²) + 1.4(3) + 2.1 = 8.73 which is 8.73% of GDP

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