To model this situation, we are going to use the decay formula:

where

is the final pupolation

is the initial population

is the Euler's constant

is the decay rate

is the time in years
A. We know for our problem that the initial population is 1,250, so

; we also know that after a year the population is 1000, so

and

. Lets replace those values in our formula to find

:







Now that we have

, we can write a function to model this scenario:

.
B. Here we are going to use a graphing utility to graph the function we derived in the previous point. Please check the attached image.
C.
- The function is decreasing
- The function doe snot have a x-intercept
- The function has a y-intercept at (0,1250)
- Since the function is decaying, it will have a maximum at t=0:



- Over the interval [0,10], the function will have a minimum at t=10:

D. To find the rate of change of the function over the interval [0,10], we are going to use the formula:

where

is the rate of change

is the function evaluated at 10

is the function evaluated at 0
We know from previous calculations that

and

, so lets replace those values in our formula to find

:



We can conclude that the rate of change of the function over the interval [0,10] is -111.572.