Find an exponential model for this data, where t = 0 corresponds to 2006. (Keep at least 5 decimal places.

Question

In the year 2006, the age-adjusted death rate per 100,000 Americans for heart disease was 241.5. In the year 2009, the age-adjusted death rate per 100,000 Americans for heart disease had changed to 234.3.

a) Find an exponential model for this data, where t = 0 corresponds to 2006. (Keep at least 5 decimal places.)

Dt=____________

b) Assuming the model remains accurate, estimate the death rate in 2034. (Round to the nearest tenth.)

_________

Answer 1

For exponential models, I prefer using the form f(x) = a(b)^x. In this form, a is the y-intercept and b is the common ratio, which we need to solve for.

In this problem, a = 241.5 and x = 3 (since 3 years have passed between 2006 and 2009)

234.3 = 241.5(b)³

.97018634 = (b)³

(.97018634)^1/3 = b

.98996 = b

D(t) = 241.5(.98996)^t

D(28) = 241.5(.98996)²⁸

D(28) = 182.1

Let me know if you have any questions or if you need to see an example with the exponential model in a different form.

Answer 2

a) D(t) = 241.5e^kt_. In 2009 t = 3. So, D(3) = 241.5e^3k = 234.3; k = ln(234.3/241.5)/3 = - 0.0103737

D(t) = 241.5e^{- 0.0103737}

b) In 2034 t = 28 and D(28) = 241.5e^{- 0.0103737· 28} = 180.62190