The national health care expenditure *H* (in billions of dollars) can be modeled (that is, accurately approximated) by the formula

*H* = 738.1(1.065)*t*

where *t* is the number of years past 1990.†

(a) What *t*-value corresponds to 2000?
*t* =  years

(b) Approximate the national health care expenditure in 2000. (Round your answer to one decimal place.)
$  billion

(c) Approximate the national health care expenditure in 2010. (Round your answer to one decimal place.)
$  billion

(d) Estimate the national health care expenditure in 2018. (Round your answer to one decimal place.)
$  billion

To solve these questions, we need to follow the given formula:

\[
H = 738.1(1.065)^t
\]

where:
- $ H $ is the national health care expenditure in billions of dollars.
- $ t $ is the number of years past 1990.

### (a) What $ t $-value corresponds to 2000?

The value of $ t $ is calculated by subtracting 1990 from the given year.

\[
t = 2000 - 1990 = 10
\]

So, $ t = 10 $ corresponds to the year 2000.

### (b) Approximate the national health care expenditure in 2000.

Using the formula:

\[
H = 738.1(1.065)^{10}
\]

Let's calculate this.

The national health care expenditure in the year 2000 is approximately \$1385.5 billion.

### (c) Approximate the national health care expenditure in 2010.

For 2010, we calculate $ t $ as:

\[
t = 2010 - 1990 = 20
\]

Using the formula, let's calculate the expenditure in 2010.

The national health care expenditure in the year 2010 is approximately \$2600.8 billion.

### (d) Estimate the national health care expenditure in 2018.

For 2018, we calculate $ t $ as:

\[
t = 2018 - 1990 = 28
\]

Using the formula, let's calculate the expenditure in 2018.

The estimated national health care expenditure in the year 2018 is approximately \$4304.3 billion.

### Summary of Results:
- **(a)** $ t = 10 $ corresponds to the year 2000.
- **(b)** The national health care expenditure in 2000 is approximately \$1385.5 billion.
- **(c)** The national health care expenditure in 2010 is approximately \$2600.8 billion.
- **(d)** The national health care expenditure in 2018 is approximately \$4304.3 billion.

Would you like more details or have any questions?

Here are 5 related questions for practice:
1. What would the national health care expenditure be in 2025 using this model?
2. How much does the national health care expenditure increase from 2000 to 2010?
3. What is the rate of change of national health care expenditure between 2010 and 2018?
4. How would you modify the formula if the growth rate changed to 1.07 instead of 1.065?
5. What would be the expenditure in 1995 based on this model?

**Tip:** When solving problems involving exponential growth, always ensure that the base and exponent are correctly applied in the formula.

Explore how to calculate the national health care expenditure using an exponential growth model based on the year. Find out the approximate expenditures for the years 2000, 2010, and 2018 using the given formula and round to one decimal place.

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