Question

1. a large mosquito infestation occurred after a recent thunderstorm in southwest florida near the beaches. the population is estimated to double in size every week. the equation ( m(t) = 15(2^t) ) can be used to determine the number of mosquitos in millions, after ( t ) weeks. calculate the average growth rate for the mosquito population for the first five weeks (0, 5).

Explanation:

Step1: Recall the average rate of change formula

The average rate of change of a function \( y = f(t) \) over the interval \([a, b]\) is given by \(\frac{f(b)-f(a)}{b - a}\).

Step2: Identify \( a \), \( b \), \( f(a) \), and \( f(b) \)

For the interval \([0, 5]\), we have \( a = 0 \), \( b = 5 \). We know \( M(t)=15(2^{t}) \), so:

• When \( t = 0 \), \( M(0)=15(2^{0})=15\times1 = 15 \) (since \( 2^{0}=1 \)).
• When \( t = 5 \), \( M(5)=15(2^{5})=15\times32 = 480 \) (since \( 2^{5}=32 \)).

Step3: Calculate the average rate of change

Using the formula \(\frac{M(5)-M(0)}{5 - 0}\), substitute the values we found:
\[
\frac{480 - 15}{5-0}=\frac{465}{5}=93
\]

Answer:

The average growth rate of the mosquito population for the first five weeks is \( 93 \) million mosquitoes per week.