The home range, in hectares, of a carnivorous mammal weighing $w$ grams can be approximated by $H(w)=0.11 w^{1.36}$. a) Find the average rate at which a carnivorous mammal's home range increases as the animal's weight grows from $300 \mathrm{~g}$ to $800 \mathrm{~g}$. b) Find $\frac{H(350)-H(250)}{350-250}$, and interpret this result.

Step 1 ：Given the function \(H(w)=0.11 w^{1.36}\), which represents the home range, in hectares, of a carnivorous mammal weighing \(w\) grams.

Step 2 ：For part a), we need to find the average rate of change of the function \(H(w)\) as \(w\) changes from \(300\) to \(800\). This can be calculated using the formula for the average rate of change, which is \(\frac{H(800)-H(300)}{800-300}\).

Step 3 ：Calculating this gives an average rate of change of approximately \(1.438\) hectares per gram.

Step 4 ：For part b), we need to find the instantaneous rate of change of the function \(H(w)\) at \(w=300\). This can be calculated using the formula for the instantaneous rate of change, which is \(\frac{H(350)-H(250)}{350-250}\).

Step 5 ：Calculating this gives an instantaneous rate of change of approximately \(1.165\) hectares per gram.

Step 6 ：Final Answer: The average rate at which a carnivorous mammal's home range increases as the animal's weight grows from \(300 \mathrm{~g}\) to \(800 \mathrm{~g}\) is approximately \(\boxed{1.438}\) hectares per gram.

Step 7 ：The value \(\frac{H(350)-H(250)}{350-250}\) is approximately \(\boxed{1.165}\), which represents the instantaneous rate of change of the home range with respect to the weight at \(w=300\) grams. This means that for a small increase in weight around \(300\) grams, the home range increases at a rate of approximately \(1.165\) hectares per gram.