Chapter 6, Problem 3CA

a.

To determine

To find : The volume of the rectangular prism formed by outer surface

Answer to Problem 3CA

The volume of the rectangular prism formed by outer surface is $T\left(x\right)=8x^3$

Explanation of Solution

Given information :

A rectangular picnic cooler with a length four times its width and height twice its width. The cooler has insulation that is 1 inch thick on each of the four sides and 2 inches thick on the top and bottom.

Let $x$ represent the width of the cooler. A polynomial function $T$ .

Calculation :

The volume of the rectangular prism formed by the outer surfaces

  $\begin{array}{l}T\left(x\right)=x\cdot 2x\cdot 4x\\ T\left(x\right)=8x^3\end{array}$

Hence, the volume of the rectangular prism formed by outer surface is $T\left(x\right)=8x^3$ .

b.

To determine

To find : The volume of the inside of the cooler

Answer to Problem 3CA

The volume of the inside of the cooler is $C\left(x\right)=8x^3-28x^2+28x-8$

Explanation of Solution

Given information :

A rectangular picnic cooler with a length four times its width and height twice its width. The cooler has insulation that is 1 inch thick on each of the four sides and 2 inches thick on the top and bottom.

A polynomial function $C$ .

Calculation :

The volume of the inside of the cooler

  $\begin{array}{l}C\left(x\right)=\left(x-2\right)\cdot \left(2x-2\right)\cdot \left(4x-2\right)\\ C\left(x\right)=\left(2x^2-4x-2x+4\right)\cdot \left(4x-2\right)\\ C\left(x\right)=\left(2x^2-6x+4\right)\cdot \left(4x-2\right)\\ C\left(x\right)=8x^3-24x^2+16x-4x^2+12x-8\\ C\left(x\right)=8x^3-28x^2+28x-8\end{array}$

Hence, the volume of the inside of the cooler is $C\left(x\right)=8x^3-28x^2+28x-8$ .

c.

To determine

To find : The volume of the insulation

Answer to Problem 3CA

The volume of the insulation is $I\left(x\right)=28x^2+28x-8$

Explanation of Solution

Given information :

A rectangular picnic cooler with a length four times its width and height twice its width. The cooler has insulation that is 1 inch thick on each of the four sides and 2 inches thick on the top and bottom.

  $I$ Be a polynomial function.

Calculation :

The volume of the insulation

  $\begin{array}{l}I\left(x\right)=T\left(x\right)-C\left(x\right)\\ I\left(x\right)=8x^3-\left(8x^3-28x^2-28x+8\right)\\ I\left(x\right)=8x^3-8x^3+28x^2+28x-8\\ I\left(x\right)=28x^2+28x-8\end{array}$

Hence, the volume of the insulation is $I\left(x\right)=28x^2+28x-8$ .

d.

To determine

To find : $I$ in standard form. What is the volume of the insulation when the width of the cooler is 8 inches?

Answer to Problem 3CA

The volume of the insulation is $I\left(8\right)=2008$ .

Explanation of Solution

Given information :

A rectangular picnic cooler with a length four times its width and height twice its width. The cooler has insulation that is 1 inch thick on each of the four sides and 2 inches thick on the top and bottom.

Calculation :

The volume of the insulation. When the width of the cooler is 8 inches.

  $\begin{array}{l}I\left(x\right)=28x^2+28x-8\\ I\left(8\right)=28\cdot {\left(8\right)}^2+28\cdot \left(8\right)-8\\ I\left(8\right)=28\cdot 64+28\cdot 8-8\\ I\left(8\right)=1792+224-8\\ I\left(8\right)=2008\end{array}$

Hence, the volume of the insulation is $I\left(8\right)=2008$ .