Fish enter a lake at a rate modeled by the function $E$ given by $E(t) = 20 + 15\sin\left(\frac{\pi t}{6}\right)$. Fish leave the lake at a rate modeled by the function $L$ given by $L(t) = 4 + 2^{0.1t^2}$. Both $E(t)$ and $L(t)$ are measured in fish per hour, and $t$ is measured in hours since midnight $(t = 0)$. (a) How many fish enter the lake over the 5-hour period from midnight $(t = 0)$ to 5 A.M. $(t = 5)$? Give your answer to the nearest whole number. (b) What is the average number of fish that leave the lake per hour over the 5-hour period from midnight $(t = 0)$ to 5 A.M. $(t = 5)$? (c) At what time $t$, for $0 \leq t \leq 8$, is the greatest number of fish in the lake? Justify your answer. (d) Is the rate of change in the number of fish in the lake increasing or decreasing at 5 a.m. $(t = 5)$? Explain your reasoning.
Fish enter a lake at a rate modeled by the function $E$ given by $E(t) = 20 + 15\sin\left(\frac{\pi t}{6}\right)$. Fish leave the lake at a rate modeled by the function $L$ given by $L(t) = 4 + 2^{0.1t^2}$. Both $E(t)$ and $L(t)$ are measured in fish per hour, and $t$ is measured in hours since midnight $(t = 0)$.
(a) How many fish enter the lake over the 5-hour period from midnight $(t = 0)$ to 5 A.M. $(t = 5)$? Give your answer to the nearest whole number.
(b) What is the average number of fish that leave the lake per hour over the 5-hour period from midnight $(t = 0)$ to 5 A.M. $(t = 5)$?
(c) At what time $t$, for $0 \leq t \leq 8$, is the greatest number of fish in the lake? Justify your answer.
(d) Is the rate of change in the number of fish in the lake increasing or decreasing at 5 a.m. $(t = 5)$? Explain your reasoning.