Chapter 3.8, Problem 36E

a.

To determine

To find: an equation that represents the given situation.

Answer to Problem 36E

  $t=\frac{k}{r}$

Explanation of Solution

Given information:

Let the time taken to empty the pool be $t$ and the rate of pumping be $r$ .

The variable $t$ varies inversely as $r$ . Let k be the constant of variation.

Calculation:

The equation for the given condition will be as follows:

  $t=\frac{k}{r}$

b.

To determine

To find: the value of $t_2$.

Answer to Problem 36E

  $36\text{ minutes}$

Explanation of Solution

Given information:

Let the time taken to empty the pool be $t$ and the rate of pumping be $r$ .

The variable $t$ varies inversely as $r$ . Let k be the constant of variation.

  $\begin{array}{l}{t}_1=45\text{ minutes}\\ r_1=800\text{ liters per minute}\\ r_2=1\text{ kiloliters per minute}=1000\text{ liters per minute}\\ t_2=?\end{array}$

Here $t_2$ is the time taken to by the new pump pumping at $r_2$ rate.

Calculation:

The equation evaluated for the given condition is shown below.

  $t=\frac{k}{r}$

Put the values in the given equation and evaluate the value of $k$ as shown below.

  $\begin{array}{c}t=\frac{k}{r}\\ k=tr\left[\text{Rearranged the equation}\right]\\ k=\left(45\right)\left(800\right)\left[\text{Put }t=t_1\text{ and }r=r_1\right]\\ =36000\end{array}$

Put the values of k and $r_2$ in the equation as follows:

  $\begin{array}{c}{t}_2=\frac{k}{r_2}\left[\text{Put }t=t_2\text{ and }r=r_2\right]\\ =\frac{36000}{1000}\\ =36\text{ minutes}\end{array}$

Thus, the time taken by new pump is 36 minutes.