Questions: A large koi pond is filled from a garden hose at the rate of 13 gal / min. Initially, the pond contains 500 gal (a) Find a linear function v that models the volume of water in the pond at any time t. V(t)= (b) If the pond has a capacity of 1722 gal, how long does it take to completely fill the pond? min

A large koi pond is filled from a garden hose at the rate of 13 gal / min. Initially, the pond contains 500 gal
(a) Find a linear function v that models the volume of water in the pond at any time t.
V(t)=

(b) If the pond has a capacity of 1722 gal, how long does it take to completely fill the pond?
min

Transcript text: A large koi pond is filled from a garden hose at the rate of $13 \mathrm{gal} / \mathrm{min}$. Initially, the pond contains 500 gal o (a) Find a linear function $v$ that models the volume of water in the pond at any time $t$. \[ V(t)= \] $\square$ (b) If the pond has a capacity of 1722 gal , how long does it take to completely fill the pond? $\square$ min

Solution

Solution Steps

Step 1: Modeling Volume Over Time

The volume of water in the pond at any time $t$ can be modeled by the linear function: $$V(t) = Rt + V_0$$ where:

$V(t)$ is the volume of water at time $t$,
$R$ is the rate of filling ($\mathrm{gal/min}$),
$V_0$ is the initial volume of water in the pond ($\mathrm{gallons}$).

Step 2: Calculating Time to Fill the Pond

To find the time $T$ required to fill the pond to its capacity, we set the volume function equal to the pond's capacity and solve for $t$: $$C = Rt + V_0$$ Rearranging the equation to solve for $t$: $$t = \frac{C - V_0}{R}$$ Substituting the given values, $C = 1722$, $V_0 = 500$, and $R = 13$, we find that: $$t = \frac{1722 - 500}{13} = 94 \, \text{minutes}$$