Chapter 7.7, Problem 48E

a.

To determine

To write: A polynomial representing the volume of the composite solid.

Answer to Problem 48E

The polynomial representing the volume of the composite solid is $x^3+16x$ .

Explanation of Solution

Given Information:A composite solid made up of a cube of side $x$

  $in.$ and a rectangular prism having sides $x$

  $in.$ ,4 $in.$ and 4 $in.$ is given.

Formulae used: Volume of cube $={(side)}^3$

Volume of rectangular prism $=(length)(width)(height)$

Calculation:

Volume of cube $=x(x)(x)$

  $=x^3$

Volume of rectangular prism $=4(x)4$

  $=16x$

Volume of composite solid= Volume of cube +Volume of rectangular prism

  $=x^3+16x$

Thus, the polynomial representing the volume of the composite solid is $x^3+16x$ .

b.

To determine

The value of $x$ when the volume of the composite solid is $25x$ .

Answer to Problem 48E

When the volume of the composite solid is $25x$ , then $x=3.$

Explanation of Solution

Given Information:A composite solid made up of a cube of side $x$

  $in.$ and a rectangular prism having sides $x$

  $in.$ ,4 $in.$ and 4 $in.$ is given. Also, its volume is given to be $25x$ .

Calculation:

From part (a), volume $=x^3+16x$

  $\Rightarrow 25x=x^3+16x$

  $\Rightarrow x^3+16x-25x=0$

  $\Rightarrow x^3-9x=0$

  $\Rightarrow x(x^2-9)=0$ (Factoring out $x$ )

  $\Rightarrow x(x+3)(x-3)=0$

  $\Rightarrow x=0$ $or$ $x=-3$ $or$ $x=3$ .

Since side cannot be zero or negative, $\therefore x=3$ .

Thus, $x=3$ , when volume is $25x$ .