Question

carl uses a pump to drain a swimming pool. the pool initially holds 105 kiloliters. assume carl can pump the water out at a constant rate of 5 kiloliters per hour. (a) find a formula for w, the number of kiloliters of water in the pool at time t. w = f(t)= (your formula contains t) (b) evaluate each. use the pull - down menu to indicate units. f(3)= number click for list f^(-1)(0)= number click for list (c) interpret the meaning of f(3). the height of the pool lowers by f(3) feet every 3 hours it is pumped. the pool loses water by f(3) kiloliters every 3 hours it is pumped. the pool contains f(3) kiloliters of water after carl has pumped for 3 hours.

Explanation:

Step1: Determine the linear - function formula

The initial amount of water is 105 kiloliters and the rate of water being pumped out is 5 kiloliters per hour. The amount of water $W$ in the pool at time $t$ (in hours) is given by the linear - function formula $W=f(t)=105 - 5t$.

Step2: Evaluate $f(3)$

Substitute $t = 3$ into the function $f(t)=105 - 5t$. So $f(3)=105-5\times3=105 - 15=90$ kiloliters.

Step3: Find the inverse of the function

First, write $y = 105 - 5t$. Solve for $t$ in terms of $y$:
$y=105 - 5t$ implies $5t=105 - y$, so $t=\frac{105 - y}{5}=21-\frac{y}{5}$. Then $f^{-1}(y)=21-\frac{y}{5}$.
Substitute $y = 0$ into $f^{-1}(y)$: $f^{-1}(0)=21-\frac{0}{5}=21$ hours.

Step4: Interpret $f(3)$

Since $f(t)$ represents the amount of water in the pool at time $t$, $f(3)$ represents the amount of water in the pool after Carl has pumped for 3 hours.

Answer:

(a) $W = f(t)=105 - 5t$
(b) $f(3)=90$ kiloliters; $f^{-1}(0)=21$ hours
(c) The pool contains $f(3)$ kiloliters of water after Carl has pumped for 3 hours.