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Q12.

Expert-verified

Algebra 1

Found in: Page 735

Book edition Student Edition

Author(s) Carter, Cuevas, Day, Holiday, Luchin

Pages 801 pages

ISBN 9780078884801

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Short Answer

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?

a. The model to represent the model is:

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

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

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

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

$l = h + 3 = (w - 1) + 3 = w + 2$ .

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 $S A = 2 (l h + h w + w l)$ , 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 $A = l h + h w + w l$

role="math" localid="1648023407780" $S A = 2 (l h + h w + w l) = 2 ((x + 2) (x - 1) + (x - 1) x + x (x + 2))$

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.

$S A = 2 (l h + h w + w l) = 2 ((x + 2) (x - 1) + (x - 1) x + x (x + 2)) = 2 (x^{2} + 2 x + x^{2} - x + x^{2} + x - 2) = 2 (3 x^{2} + 2 x - 2)$

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

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 $S A = 2 (l h + h w + w l)$ , 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 localid="1648023988206" $A = l h + h w + w l$ .

$S A = 2 (l h + h w + w l) = 2 ((x + 2) (x - 1) + (x - 1) x + x (x + 2)) = 2 (x^{2} + 2 x + x^{2} - x + x^{2} + x - 2) = 2 (3 x^{2} + 2 x - 2)$

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

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.

$2 (3 x^{2} + 2 x - 2) = 62 3 x^{2} + 2 x - 2 = 31 3 x^{2} + 2 x - 33 = 0 3 x^{2} + 11 x - 9 x - 33 = 0 x (3 x + 11) - 3 (3 x + 11) = 0 (x - 3) (3 x + 11) = 0$

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.

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