Chapter 9.1, Problem 52PPS

a.

To determine

Find the length of $BC$ in terms of $x$ .

Answer to Problem 52PPS

The length of $BC=\frac{20}{x}$ centimeters

Explanation of Solution

Calculation:

The area $A$ of a rectangle is given by the formula $A=lw\text{ where }l=\text{length and }w=\text{width}$

Use the formula above with $A=20$ square centimeters and assume that the width is $AB=x$ to obtain:

  $\begin{array}{l}A=lw\\ =\left(BC\right)\left(x\right)\\ \frac{20}{x}=\frac{\left(BC\right)\left(x\right)}{x}\\ \frac{20}{x}=BC\end{array}$

  $BC=\frac{20}{x}$ Centimeters.

Thus, the length of $BC=\frac{20}{x}$ centimeters.

b.

To determine

Determine the area of the shade region in terms of $x$ .

Answer to Problem 52PPS

Thearea of the region is $A=\frac{1200}{x^2}\text{squarecm}$

Explanation of Solution

Calculation:

From part (a) $BC=\frac{20}{x}$

If $DC=3BC$ ,then

  $\begin{array}{l}DC=3\left(BC\right)\\ =3\left(\frac{20}{x}\right)\\ =\frac{60}{x}cm\end{array}$

The shade region, which is a rectangle, has $BC=\frac{20}{x}\text{and}DC=\frac{60}{x}.$

The area is:

  $\begin{array}{l}A=lw\\ =\left(BC\right)\left(DC\right)\\ A=\frac{20}{x}\cdot \frac{60}{x}\\ A=\frac{1200}{x^2}\text{squarecm}\end{array}$

Thus, the area of the region is $A=\frac{1200}{x^2}\text{squarecm}$

c.

To determine

Determine the volume of the region in terms of $x$ .

Answer to Problem 52PPS

Thevolume of the prism in terms of $\frac{1200}{x}\text{cubic cm}\text{.}$

Explanation of Solution

Calculation:

The volume V of a rectangular prism is given by the formula

  $V=lwh,\text{where }l=\text{length},w=\text{width},\text{and}h=\text{height}.$

The given rectangular prism has:

  $\begin{array}{l}l=BC=\frac{20}{x}\text{cm}\\ w=AB=x\text{cm}\\ h=DC=\frac{60}{x}\text{cm}\end{array}$

Use the formula above to obtain:

  $\begin{array}{l}V=lwh\\ =\left(BC\right)\left(AB\right)\left(DC\right)\\ =\frac{20}{x}\cdot x\cdot \frac{60}{x}\\ =\frac{1200x}{x^2}\\ =\frac{1200}{x}\text{cubic cm}\text{.}\end{array}$

Thus, the volume of the prism in terms of $\frac{1200}{x}\text{cubic cm}\text{.}$