Question

4. The area covered by a lake is 11 square kilometers. It is decreasing exponentially at a

rate of 2 percent each year and can be modeled by A(t) = 11 -0.98¹.
a. By what factor does the area decrease in 10 years?
b. By what factor does the area decrease each month?

Answer 1

a. To find the factor by which the area decreases in 10 years, we can use the formula for exponential decay: A(t) = A0 * e^(-kt) where A0 is the initial area, k is the decay constant, and t is the time in years.

Given that A(t) = 11 -0.98^t.

so, we can say k =0.98

The area after 10 years will be:

A(10) = 11 - 0.98^10

The decrease factor is:

(A(10)/A(0)) = (11 - 0.98^10)/11

b. To find the factor by which the area decreases each month, we can first convert the annual decay rate to a monthly rate by dividing by 12, since there are 12 months in a year.

So, the monthly decay rate is 0.98/12 = 0.0816

Then we can use the formula for exponential decay again with the new decay rate:

A(t) = A0 * e^(-kt)

The area after 1 month will be:

A(1/12) = 11 - 0.0816^(1/12)

The decrease factor is :

(A(1/12)/A(0)) = (11 - 0.0816^(1/12))/11

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