Question

question 5
a mouse population, m, can be estimated with the function
m(t) = \frac{203}{1+4e^{-0.3t}}
where t is measured in years and t = 0 represents the mouse population in the year 2015.
what is the average rate of change of the mouse population from 2020 to 2025?

a. the population increases at a rate of 12.40 mice per year.
b. the population increases at a rate of 1.54 mice per year.
c. the population increases at a rate of 62.02 mice per year.
d. the population increases at a rate of 0.31 mice per year

Explanation:

Step1: Determine t values for 2020 and 2025

Since \( t = 0 \) is 2015, for 2020, \( t=2020 - 2015=5 \). For 2025, \( t = 2025 - 2015 = 10 \).

Step2: Calculate \( M(5) \)

Substitute \( t = 5 \) into \( M(t)=\frac{203}{1 + 4e^{-0.3t}} \):
\( M(5)=\frac{203}{1+4e^{-0.3\times5}}=\frac{203}{1 + 4e^{-1.5}} \)
\( e^{-1.5}\approx0.2231 \), so \( 4e^{-1.5}\approx0.8924 \)
\( M(5)=\frac{203}{1 + 0.8924}=\frac{203}{1.8924}\approx107.27 \)

Step3: Calculate \( M(10) \)

Substitute \( t = 10 \) into \( M(t) \):
\( M(10)=\frac{203}{1+4e^{-0.3\times10}}=\frac{203}{1 + 4e^{-3}} \)
\( e^{-3}\approx0.0498 \), so \( 4e^{-3}\approx0.1992 \)
\( M(10)=\frac{203}{1+0.1992}=\frac{203}{1.1992}\approx169.27 \)

Step4: Calculate average rate of change

The average rate of change formula is \( \frac{M(10)-M(5)}{10 - 5} \)
\( \frac{169.27-107.27}{5}=\frac{62}{5} = 12.4 \)

Answer:

A. The population increases at a rate of 12.40 mice per year.