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 $L ( t ) = 4 + 2 ^ { 0.1 t ^ { 2 } }$. 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 $L ( t ) = 4 + 2 ^ { 0.1 t ^ { 2 } }$. 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.