Q12. - Algebra 1

Question

Carl’s father is building a tool chest that is shaped like a rectangular prism. He wants the tool chest to have a surface area of 62 square feet. The height of the chest will be 1 foot shorter than the width. The length will be 3 feet longer than the height.

 

a. Sketch the model to represent the model.

b. Write a polynomial that represents the surface area of the tool chest.

c. What are the dimensions of the tool chest?

Step-by-Step Solution

Answer

a. The model to represent the model is:

b. The polynomial that represents the surface area of the tool chest is $2\left(3x^2+2x-2\right)$. 

c. The dimensions of the tool chest are Length is 5 feet, height is 2 feet and width is 3 feet.

Part a. Step 1. Assign the variables.

Let the length of the rectangular prism be $l$, width be $w$ and height be $h$.

Part a. Step 2. Represent the variables in terms of width.

Since height is 1 foot shorter than width, therefore, the height of rectangular prism in terms of width is $h=w-1$. Similarly, length is 3 feet longer than height, therefore, the length of a rectangular prism in terms of width is

$$\begin{gathered} l=h+3 \\ =\left(w-1\right)+3 \\ =w+2 \end{gathered}$$.

Part a. Step 3. Sketch the model.

Draw the figure in terms of width.

Part b. Step 1. Define the formula for the surface area of a cuboid.

The formula for the surface area of a rectangular prism is given by $SA=2\left(lh+hw+wl\right)$, where $l$ is the length, $h$ is the height, and $w$ is the width.

Part b. Step 2. Substitute the values.

Let $x$ be the width, then the height will be $x-1$ and length will be $x+2$. Substitute these values of $w, h$ and $l$ into the equation $A=lh+hw+wl$

$$\begin{gathered} SA=2\left(lh+hw+wl\right) \\ =2\left(\left(x+2\right)\left(x-1\right)+\left(x-1\right)x+x\left(x+2\right)\right) \end{gathered}$$

Part b. Step 3. Simplify the expression.

Simplify the equation in order to find the polynomial that represents the surface area of the tool chest.

 $$\begin{gathered} SA=2\left(lh+hw+wl\right) \\ =2\left(\left(x+2\right)\left(x-1\right)+\left(x-1\right)x+x\left(x+2\right)\right) \\ =2\left(x^2+2x+x^2-x+x^2+x-2\right) \\ =2\left(3x^2+2x-2\right) \end{gathered}$$ 

The polynomial that represents the surface area of the tool chest is $2\left(3x^2+2x-2\right)$. 

Part c. Step 1. Define the formula for the surface area of a cuboid.

The formula for the surface area of a rectangular prism is given by $SA=2\left(lh+hw+wl\right)$, where $l$ is the length, $h$ is the height, and $w$ is the width.

Part c. Step 2. Find the polynomial that represents the surface area of the tool chest.

Let $x$ be the width, then the height will be $x-1$ and length will be $x+2$. Substitute these values of $w, h$ and $l$ into the equation $A=lh+hw+wl$.

$$\begin{gathered} SA=2\left(lh+hw+wl\right) \\ =2\left(\left(x+2\right)\left(x-1\right)+\left(x-1\right)x+x\left(x+2\right)\right) \\ =2\left(x^2+2x+x^2-x+x^2+x-2\right) \\ =2\left(3x^2+2x-2\right) \end{gathered}$$

The polynomial that represents the surface area of the tool chest is $2\left(3x^2+2x-2\right)$. 

Part c. Step 3. Find the dimensions of the tool chest.

It is given that the surface area of the tool chest is 62 square feet. 

$$\begin{gathered} 2\left(3x^2+2x-2\right)=62 \\ 3x^2+2x-2=31 \\ 3x^2+2x-33=0 \\ 3x^2+11x-9x-33=0 \\ x\left(3x+11\right)-3\left(3x+11\right)=0 \\ \left(x-3\right)\left(3x+11\right)=0 \end{gathered}$$

Therefore, $x=3,\frac{-11}{3}$. Since width cannot be negative, therefore, $x=3$ is the valid value.

So, Length = $x+2=5$

And, Height = $x-1=2$.

Therefore, the dimensions of the tool chest are Length is 5 feet, height is 2 feet and width is 3 feet.