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A population of bees is decreasing. The population in a particular region this year is 1,250. After 1 year, it is estimated that the population will be 1,000. After 3 years, it is estimated that the population will be 640. a. Write a function to model this scenario. b. Create a graph to show the bee population over the next 10 years. c. Identify the key features of the function. Identify the x- and y-intercepts. Determine the maximum, the minimum, whether the function is increasing or decreasing, the rate of change of the function over the interval [0, 10], and any asymptotes.

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To model this situation, we are going to use the decay formula:
A=Pe^(rt)
where

A is the final pupolation

P is the initial population

e is the Euler's constant

r is the decay rate

t is the time in years

A. We know for our problem that the initial population is 1,250, so
P=1250; we also know that after a year the population is 1000, so
A=1000 and
t=1. Lets replace those values in our formula to find
r:

A=Pe^(rt)

1000=1250e^(r)

e^(r)= (1000)/(1250)

e^(r)= (4)/(5)

ln(e^(r))=ln( (4)/(5) )

r=ln( (4)/(5) )

r=-02231

Now that we have
r, we can write a function to model this scenario:

A(t)=1250e^(-0.2231t).

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:

A(0)=1250e^{(-0.2231)(0)

A_(0)=1250e^(0)

A_(0)=1250
- Over the interval [0,10], the function will have a minimum at t=10:

A(10)=1250e^{(-0.2231)(10)

A_(10)=134.28

D. To find the rate of change of the function over the interval [0,10], we are going to use the formula:
m= (A(0)-A(10))/(10-0)
where

m is the rate of change

A(10) is the function evaluated at 10

A(0) is the function evaluated at 0
We know from previous calculations that
A(10)=134.28 and
A(0)=1250, so lets replace those values in our formula to find
m:

m= (134.28-1250)/(10-0)

m= (-1115.72)/(10)

m=-111.572
We can conclude that the rate of change of the function over the interval [0,10] is -111.572.
 A population of bees is decreasing. The population in a particular region this year-example-1
User Tajuddin Khandaker
by
8.3k points

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