Q22. - Algebra 1

Question

A rectangular prism has dimensions $\mathit{x}$, ,$\mathit{x}\mathbf{+}\mathbf{3}$ and $\mathbf{2}\mathit{x}\mathbf{+}\mathbf{5}$.

a. Find the volume of the prism in terms of $\mathit{x}$.

b. Choose two values for $\mathit{x}$. How do the volumes compare?

Step-by-Step Solution

Answer

a) The volume of the prism in terms of $x$ is $\mathbf{2}{\mathit{x}}^{\mathbf{3}}\mathbf{+}\mathbf{11}{\mathit{x}}^{\mathbf{2}}\mathbf{+}\mathbf{15}\mathit{x}$.

b) Let the two values $x$ of be 1 and 2. The volume when $x=1$ is 28 and the volume when $x=2$ is 90. The volume gets increased on increasing the value of $\mathit{x}$.

Part a Step 1. Write the formula for the volume of the rectangular prism.

The volume ($V$) of the rectangular prism is given by:

$V=L\times B\times H$

Where $L, B$ and $H$ are the length, breadth and height of the rectangular prism respectively.

Part a Step 2. Find the volume of the prism in terms of x .

Substitute, $x+3$ and $2x+5$ for $L, B$ and $H$ in the formula for the volume of the rectangular prism.

Therefore, it is obtained that:

$\begin{array}{l}V=L\times B\times H\\ =x(x+3)(2x+5)\\ =(x^2+3x)(2x+5)\\ =2x^3+5x^2+6x^2+15x\\ =2x^3+11x^2+15x\end{array}$

Therefore, the volume of the prism in terms of $x$ is $2x^3+11x^2+15x$.

Part b Step 1. Choose two values for x .

Let the two values of $x$ be 1 and 2.

Part b Step 2. Find the volumes when x = 1 and x = 2 .

The volume of the prism in terms of $x$ is$2x^3+11x^2+15x$.

Substitute 1 for $x$ and find the volume.

$\begin{array}{l}V=2x^3+11x^2+15x\\ =2{\left(1\right)}^3+11{\left(1\right)}^2+15\left(1\right)\\ =2+11+15\\ =28\end{array}$

Substitute 2 for $x$ and find the volume.

$\begin{array}{l}V=2x^3+11x^2+15x\\ =2{\left(2\right)}^3+11{\left(2\right)}^2+15\left(2\right)\\ =16+44+30\\ =90\end{array}$

Therefore, it can be noticed that the volume gets increased on increasing the value of $x$.