Chapter 1.5, Problem 86E

a.

To determine

To find:the function that gives the area of the circular leak in terms of the time.

Answer to Problem 86E

The value of function that gives the area of the circular leak in terms of time is $27.56t\pi$.

Explanation of Solution

Given information :

The functions are $r\left(t\right)=5.25\sqrt{t}$

Calculation : since the area of a circle is $a=\pi r^2$ ,to find the composite function substitute $r$ by the function $r(t)$ ,

  $\begin{array}{l}A=a\left(5.25\sqrt{t}\right)\text{ [}r\left(t\right)=5.25\sqrt{t}]\\ A=\pi {\left(5.25\sqrt{t}\right)}^2\text{ [a}\left(r\right)=\pi r^2]\\ A=\pi \left(27.56t\right)\text{ [expand]}\\ A=27.56t\pi \text{ [multiply]}\end{array}$

Thus, The value of function that gives the area of the circular leak in terms of time is $27.56t\pi$.

b.

To determine

To find: the size of the contaminated area.

Answer to Problem 86E

The size of the contaminated area is $992.16\pi {\text{ m}}^2$.

Explanation of Solution

Given information :

The function is $r\left(t\right)=5.25\sqrt{t}$.

Calculation : to find size of the contaminated area after $36$ hours,

since area is $27.56t\pi$ , substitute $t=36$ ,

  $\begin{array}{l}A=27.56t\pi \\ A=27.56\left(36\right)\pi \text{ [}t=36]\\ A=992.16\pi {\text{ m}}^2\text{ [multiply]}\end{array}$

Thus, The size of the contaminated area is $992.16\pi {\text{ m}}^2$.

c.

To determine

To find: the time when size is $6250{\text{ m}}^2$ .

Answer to Problem 86E

The time is $72.22$ hours when size is $6250{\text{ m}}^2$.

Explanation of Solution

Given information :

The function is $r\left(t\right)=5.25\sqrt{t}$.

Calculation : to find the time substitute $A=6250$ ,

The value of function $A$ is $27.56t\pi$ ,

  $\begin{array}{l}A=27.56t\pi \\ 27.56t\pi =6250\text{ [substitute]}\\ t=72.22\text{ [simplify]}\end{array}$

Thus, The time is $72.22$ hours when size is $6250{\text{ m}}^2$.