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).