Free Response Using Accumulation and Definite Integral in Applied Contexts $($Rate In$/$Rate Out Problem$)$

$($calculator allowed$)$

A popular $24-$hour health club, Get Swole, has $29$ people using its facility at time $t=0.$ During the time interval $0t20$ hours, people are entering the health club at the rate

$E(t)=-0\mathrm{.}018t^2+11$ people per hour.

During the same time period people are leaving the health club at the rate of

$L(t)=0\mathrm{.}013t^2-0\mathrm{.}25t+8$ people per hour.

a$.)$ Is the number of people in the facility increasing or decreasing at time $t=11?$ Explain your reasoning.

$I(t)=-\mathrm{.}018t^2+11-\mathrm{.}013t^2-\mathrm{.}25t+8$

Decrosing because the

derivitove of the number

at fecele in the facilty.

$I^{\text{'}}(1)=-\mathrm{.}932$

of people in the facillity is negative,

b$.)$ To the nearest whole number, how many people are in the health club at time $t=20.$

c$.)$ At what time $t,$ for $0t20,$ is the amount of people in the health club a maximum? Justify your answer.

d$.)$ If the health club has less than $10$ people using its facility, the owner will serve everyone a special health drink. Set up$,$ but do not solve, an inequality involving one or more integrals that could be used to find the time, $H,$ when the number of people in the health club is less than or equal to $10.$

$14$