Question

In the year 1998, the age-adjusted death rate per 100,000 Americans from heart disease was 248.2. In the year 2004, the age-adjusted death rate per 100,000 Americans for heart disease had changed to 213.2.

a) Find an exponential model for this data, where t=0 corresponds to 1998.

b) Assuming the model remains accurate, estimate the death rate in 2025. (Round to the nearest tenth.)

Answer 1

A: y = 248.2 * (0.9896)^t

B: 169.6

a) To find an exponential model for this data, we can use the formula:

y = ab^t

where y is the age-adjusted death rate per 100,000 Americans from heart disease, a is the initial death rate, b is the growth factor, and t is the number of years since 1998.

We are given two data points:

(0, 248.2) for 1998

(6, 213.2) for 2004

Let's plug in the first data point to find 'a':

248.2 = a * b^0

Since any number raised to the power of 0 is 1, we have:

a = 248.2

Now let's plug in the second data point and 'a' to find 'b':

213.2 = 248.2 * b^6

To find 'b', we'll first divide both sides by 248.2:

213.2 / 248.2 = b^6

0.859033 = b^6

Now take the sixth root of both sides to solve for 'b':

b = (0.859033)^(1/6)

b ≈ 0.9896

Our exponential model is:

y = 248.2 * (0.9896)^t

b) To estimate the death rate in 2025, we need to find the value of 't' for 2025:

t = 2025 - 1998 = 27

Now, we can plug 't' into our exponential model:

y = 248.2 * (0.9896)^27

y ≈ 169.6

Assuming the model remains accurate, the estimated age-adjusted death rate per 100,000 Americans from heart disease in 2025 is approximately 169.6 (rounded to the nearest tenth).