Chapter 4.4, Problem 49E

Arctic Foxes The age-weight relationship of female Arctic foxes caught in Svalbard, Norway, can be estimated by the function

$M\left(t\right)=3102e^{-e^{-0.022\left(t-56\right)}},$

where $t$ is the age of the fox in days and $M\left(t\right)$ is the weight of the fox in grams. Source: Journal of Mammalogy.

a. Estimate the weight of a female fox that is $200$ days old.

b. Use $M\left(t\right)$ to estimate the largest size that a female fox can attain. (Hint: Find $\underset{t\to \infty }{\lim }M\left(t\right)$.)

c. Estimate the age of a female fox when it has reached $80\%$ of its maximum weight.

d. Estimate the rate of change in weight of an Arctic fox that is $200$ days old. (Hint: Recall that $D_t\left[e^{f\left(t\right)}\right]=f\text{'}\left(t\right)e^{f\left(t\right)}.)$

e. Using a graphing calculator to graph $M\left(t\right)$ and then describe the growth pattern.

f. Use the table function on a graphing calculator or a spreadsheet to develop a chart that shows the estimated weight and growth rate of female foxes for days $50,100,150,200,250,$ and $300$.