Chapter 1, Problem 13CT

To determine

To describe: The way to transform the graph for second fountain and then to find depth of each fountain after $4$ minutes.

Answer to Problem 13CT

When $x=4\min .$ , for first fountain depth is $4feet$ , that for second fountain is $8feet$ .

Explanation of Solution

Given information: Function representing the first fountain is $f(t)=\frac{1}{4}\left|t-20\right|$ .

Calculation:

As the second fountains gets refilled twice as quickly as first, the water level should be doubled for the same $t$ values. This implies that whole function should be multiplied by $2$ , therefore, there should be a vertical stretch by a factor of $2$ ,

For second fountain,

  $\begin{array}{l}g(t)=2f(t)\\ g(t)=\frac{1}{2}\left|t-20\right|\end{array}$

When $x=4$ for first fountain,

  $\begin{array}{l}f(4)=\frac{1}{4}\left|4-20\right|\\ =\frac{1}{4}\times 16\\ =4\end{array}$

When $x=4$ for second fountain,

  $\begin{array}{l}g(4)=\frac{1}{2}\left|4-20\right|\\ =\frac{1}{2}\times 16\\ \\ =8\end{array}$

As $\frac{Depthof2ndfountain}{Depthof1stfountain}=\frac{8}{4}=2$ , the function is correct,

Thus, when $x=4\min .$ , for first fountain depth is $4feet$ , that for second fountain is $8feet$ .