Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![### Word Problem: Wrapping Paper Calculation
**Problem Statement:**
Susan bought two gifts. One package is a rectangular prism with a base length of 4 inches, a base width of 2 inches, and a height of 10 inches. The other package is a cube with a side length of 5 inches. Which package requires more wrapping paper to cover? What is the total amount of wrapping paper Susan must use to cover both packages? You must show your work to earn full credit.
**Solution:**
To determine which package requires more wrapping paper, we need to calculate the surface area of both the rectangular prism and the cube.
**Step 1: Calculate the Surface Area of the Rectangular Prism**
The surface area \( SA \) of a rectangular prism is given by the formula:
\[ SA = 2lw + 2lh + 2wh \]
where \( l \) is the length, \( w \) is the width, and \( h \) is the height.
For the given rectangular prism:
- Length \( l = 4 \) inches
- Width \( w = 2 \) inches
- Height \( h = 10 \) inches
\[ SA = 2(4 \times 2) + 2(4 \times 10) + 2(2 \times 10) \]
\[ SA = 2(8) + 2(40) + 2(20) \]
\[ SA = 16 + 80 + 40 \]
\[ SA = 136 \text{ square inches} \]
**Step 2: Calculate the Surface Area of the Cube**
The surface area \( SA \) of a cube is given by the formula:
\[ SA = 6s^2 \]
where \( s \) is the side length.
For the given cube:
- Side length \( s = 5 \) inches
\[ SA = 6(5)^2 \]
\[ SA = 6(25) \]
\[ SA = 150 \text{ square inches} \]
**Comparison:**
- Rectangular prism surface area = 136 square inches
- Cube surface area = 150 square inches
The cube requires more wrapping paper than the rectangular prism.
**Step 3: Calculate the Total Amount of Wrapping Paper Needed**
Total surface area required:
\[ \text{Total Surface Area} = 136 \text{ square inches} +](https://content.bartleby.com/qna-images/question/1aebde78-3dd9-49e5-ba05-c4ce5860123f/e294afc3-55bd-4b37-848a-e4abe96c4f93/obm81ra_processed.jpeg)
Transcribed Image Text:### Word Problem: Wrapping Paper Calculation
**Problem Statement:**
Susan bought two gifts. One package is a rectangular prism with a base length of 4 inches, a base width of 2 inches, and a height of 10 inches. The other package is a cube with a side length of 5 inches. Which package requires more wrapping paper to cover? What is the total amount of wrapping paper Susan must use to cover both packages? You must show your work to earn full credit.
**Solution:**
To determine which package requires more wrapping paper, we need to calculate the surface area of both the rectangular prism and the cube.
**Step 1: Calculate the Surface Area of the Rectangular Prism**
The surface area \( SA \) of a rectangular prism is given by the formula:
\[ SA = 2lw + 2lh + 2wh \]
where \( l \) is the length, \( w \) is the width, and \( h \) is the height.
For the given rectangular prism:
- Length \( l = 4 \) inches
- Width \( w = 2 \) inches
- Height \( h = 10 \) inches
\[ SA = 2(4 \times 2) + 2(4 \times 10) + 2(2 \times 10) \]
\[ SA = 2(8) + 2(40) + 2(20) \]
\[ SA = 16 + 80 + 40 \]
\[ SA = 136 \text{ square inches} \]
**Step 2: Calculate the Surface Area of the Cube**
The surface area \( SA \) of a cube is given by the formula:
\[ SA = 6s^2 \]
where \( s \) is the side length.
For the given cube:
- Side length \( s = 5 \) inches
\[ SA = 6(5)^2 \]
\[ SA = 6(25) \]
\[ SA = 150 \text{ square inches} \]
**Comparison:**
- Rectangular prism surface area = 136 square inches
- Cube surface area = 150 square inches
The cube requires more wrapping paper than the rectangular prism.
**Step 3: Calculate the Total Amount of Wrapping Paper Needed**
Total surface area required:
\[ \text{Total Surface Area} = 136 \text{ square inches} +
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