The national health expenditures are projected to grow at...

The national health expenditures are projected to grow at the rate of $$ r(t)=0.0058 t+0.159 \quad(0 \leq t \leq 13) $$ trillion dollars/year from 2002 through $2015 .$ Here, $t=0$ corresponds to 2002. The expenditure in 2002 was $$\$ 1.60$$ trillion. a. Find a function $f$ giving the projected national health expenditures in year $t$. b. What does your model project the national health expenditure to be in 2015 ?

The national health expenditures are projected to grow at the rate of
$$
r(t)=0.0058 t+0.159 \quad(0 \leq t \leq 13)
$$
trillion dollars/year from 2002 through $2015 .$ Here, $t=0$ corresponds to 2002. The expenditure in 2002 was $$\$ 1.60$$ trillion.
a. Find a function $f$ giving the projected national health expenditures in year $t$.
b. What does your model project the national health expenditure to be in 2015 ?

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Key Concepts

Rate of Change

A rate of change function expresses how a quantity changes with respect to time. It provides the instantaneous rate at which a variable, such as national health expenditures, increases or decreases over time.

Antiderivatives and Integration

Integration is the process of finding an antiderivative, which allows one to accumulate the effects of a rate function over time. This process transforms a rate of change function into a total accumulated quantity by summing up all the individual changes.

Initial Value Problems

When integrating a rate function to model a real-world quantity, an initial condition (starting value) is used to determine the constant of integration. This ensures that the resulting model accurately reflects the known state at a specific time.

Mathematical Modeling

Mathematical modeling involves creating functions that represent real-world phenomena, such as future national health expenditures. It combines rate functions, integration, and initial conditions to project future values based on current data.

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