Chapter 8.1, Problem 12CYU

a.

To determine

To find the length and width of the container.

Answer to Problem 12CYU

The height, length and width of the container is $x,\left(2x+1\right)\text{ and }\left(3x+4\right)$

Explanation of Solution

Given: The volume of a shipping container in the shape of a rectangular prism is given by the polynomial: $6x^3+11x^2+4x$

Calculation:

The volume of the container is given as:

  $\begin{array}{l}6x^3+11x^2+4x\\ =x\left(6x^2+11x+4\right)\\ =x\left(6x^2+3x+8x+4\right)\\ =x\left[3x\left(2x+1\right)+4\left(2x+1\right)\right]\\ =x\left(2x+1\right)\left(3x+4\right)\end{array}$ [Factorize and simplify the expression]

Hence,

The height, length and width of the container is $x,\left(2x+1\right)\text{ and }\left(3x+4\right)$

b.

To determine

To find the ratio of the three dimensions of the container when $x=2.$

Answer to Problem 12CYU

The ratio of the dimensions of the container $2:5:10.$

Explanation of Solution

Given: height $x=2.$

Calculation:

From part (a), Height $x$ , length $2x+1$ ,width is $3x+4.$

When $x=2$ , the height will be equal to $2$

Similarly,

Length of container is $2x+1$ ,

  $\begin{array}{l}2\left(2\right)+1=4+1\text{ substitute }x=2\text{ in length}\\ =5\end{array}$

Width of container is $3x+4$ ,

  $\begin{array}{l}3\left(2\right)+4=6+4\text{ substitute }x=2\text{ in width}\\ =10\end{array}$

Thus, the ratio of the dimensions of the container is $2:5:10.$

c.

To determine

To find the all values of $x$ .

Answer to Problem 12CYU

The ratio of the dimensions of the container is $3:7:13.$

Explanation of Solution

Given: From part (a), Height $x$ , length $2x+1$ , width is $3x+4.$

Calculation:

Height $x$ , length $2x+1$ ,width is $3x+4$

Evaluate each dimension when x is equal to value other than 2.

Let Height is $x=3$ .

Length of container is $2x+1$ ,

  $\begin{array}{l}2\left(3\right)+1=6+1\text{ substitute }x=3\text{ in length}\\ =7\end{array}$

Width of container is $3x+4$ ,

  $\begin{array}{l}3\left(3\right)+4=9+4\text{ substitute }x=3\text{ in width}\\ =13\end{array}$

Ratio of the dimensions of the container is $3:7:13.$

Thus, the ratio of dimensions of the container will not be the same for each $x$ value.