Based on the current eligibility requirement, a study...

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 $$ P(t)=0.27 t^{2}+1.4 t+2.2 \quad(0 \leq t \leq 5) $$ percent in year $t,$ where $t$ is measured in decades, with $t=0$ corresponding to $2000 .$ a. How fast was the spending on Medicare, as a percentage of the GDP, growing in $2010 ?$ How fast will it be growing in $2020 ?$ b. What was the predicted spending on Medicare in $2010 ?$ What will it be in $2020 ?$ Source: Congressional Budget Office.

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
$$
P(t)=0.27 t^{2}+1.4 t+2.2 \quad(0 \leq t \leq 5)
$$
percent in year $t,$ where $t$ is measured in decades, with $t=0$ corresponding to $2000 .$
a. How fast was the spending on Medicare, as a percentage of the GDP, growing in $2010 ?$ How fast will it be growing in $2020 ?$
b. What was the predicted spending on Medicare in $2010 ?$ What will it be in $2020 ?$
Source: Congressional Budget Office.

Key Concepts

Polynomial Functions

Polynomial functions, made up of terms with non-negative integer exponents, are widely used to model real-world situations. Understanding their structure facilitates both the process of differentiation and evaluation, which are key in predicting trends such as growth in Medicare spending.

Unit Conversion

Unit conversion is the process of ensuring that all units in a problem are consistent. In this problem, time is measured in decades, so converting calendar years (like 2010 and 2020) into the corresponding values of t in the function is essential for correct calculation.

Differentiation

Differentiation is the process of finding the derivative of a function, which represents the rate of change of the function with respect to its variable. In this context, differentiating the spending function helps determine how fast Medicare spending is growing at specific future years.

Function Evaluation

Function evaluation involves substituting specific values into a function to obtain the corresponding output. Here, the spending function is evaluated at particular time points to predict the percentage of GDP spent on Medicare in future years.