Question

Filling a Pond A large koi pond is filled from a garden hose at the rate of $\text{10gal}/\text{min}$. Initially, the pond contains$300\text{gal}$ of water.

(a) Find a linear function V that models the volume of water in the pond at any time t.

(b) If the pond has a capacity of $1300\text{gal}$, how long does it take to completely fill the pond?

Short answer

(a) The linear function that models the volume of water in the pond is given by $V\left(t\right)=10t+300$.

(b) The pond is filled completely in$100$ minutes if the capacity of pond is $1300\text{gal}$.

Step-by-step solution

Part a. Step 1. List out the given information.

The rate at which the pond is filled is $\text{10gal}/\text{min}$.

The amount of water in the pond initially is $300\text{gal}$.

Part a. Step 2. Apply the concept of linear function.

Let a function that model the volume of water in the pond at any time t be,

$V\left(t\right)=at+b\mathrm{...}\left(1\right)$

Here, a is the slope of the equation and b is the constant or value at initial condition.

The rate at which the water is filled is the slope.

So,

$\begin{array}{l}a=10\\ b=300\end{array}$

Part a. Step 3. Substitute the values of a and b.

Substitute the values of a andb in equation (1).

$V\left(t\right)=10t+300$

Thus, the linear function that models the volume of water in the pond is given by $V\left(t\right)=10t+300$.

Part b. Step 1. List the given information.

The volume of water in pond is $1300\text{gal}$.

$V\left(t\right)=1300\text{gal}$

Part b. Step 2. Write the equation of model.

The linear function that models the volume of water in the pond is given by,

$V\left(t\right)=10t+300\mathrm{...}\left(2\right)$

Here,$V\left(t\right)$ is the volume of water in the pond and t is the time.

Part b. Step 3. Find the value of t.

Substitute the value of t in equation (2).

$$\begin{gathered} 1300=10t+300 \\ 10t=1300-300 \\ t=\frac{1000}{10} \\ =100\text{minutes} \end{gathered}$$

Thus, the pond is filled completely in$100$minutes if the capacity of pond is $1300\text{gal}$.