4. If a cardboard box (rectangular prism) with a top, a bottom, and 4 sides is W ft wide, L ft long, and H ft tall, then what is the surface area of this box? Find a formula for the surface area in terms of W, L, and H. Explain clearly why your formula is valid.
4. If a cardboard box (rectangular prism) with a top, a bottom, and 4 sides is W ft wide, L ft long, and H ft tall, then what is the surface area of this box? Find a formula for the surface area in terms of W, L, and H. Explain clearly why your formula is valid.
Elementary Algebra
17th Edition
ISBN:9780998625713
Author:Lynn Marecek, MaryAnne Anthony-Smith
Publisher:Lynn Marecek, MaryAnne Anthony-Smith
Chapter8: Rational Expressions And Equations
Section8.9: Use Direct And Inverse Variation
Problem 508E: The area of the face of a Ferris wheel varies directly with the square of its radius. If the area of...
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![**Problem 4:**
If a cardboard box (rectangular prism) with a top, a bottom, and 4 sides is \( W \) ft wide, \( L \) ft long, and \( H \) ft tall, then what is the surface area of this box? Find a formula for the surface area in terms of \( W \), \( L \), and \( H \). Explain clearly why your formula is valid.
**Explanation:**
To find the surface area of a rectangular prism (a cardboard box in this case), you need to calculate the area of all six faces and then sum them up.
A rectangular prism has three pairs of identical opposing faces:
1. The top and bottom faces.
2. The front and back faces.
3. The left and right side faces.
- The top and bottom faces each have an area of \( L \times W \).
- The front and back faces each have an area of \( L \times H \).
- The left and right side faces each have an area of \( W \times H \).
Thus, the total surface area \( A \) is calculated as follows:
\[ A = 2(L \times W) + 2(L \times H) + 2(W \times H) \]
This simplifies to:
\[ A = 2LW + 2LH + 2WH \]
This formula is valid because it correctly sums the areas of all six faces of the rectangular prism. Each dimension combination (\( LW \), \( LH \), and \( WH \)) appears twice in the process, corresponding to their pair of identical faces. This ensures that all surface areas of the box are included in the total calculation.
Therefore, the surface area \( A \) of the rectangular box in terms of \( W \), \( L \), and \( H \) is given by:
\[ A = 2(LW + LH + WH) \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7c60e9dc-9541-469d-8098-00c54f69ea68%2F98700987-70d6-48de-ab68-628181731860%2Ftxnpdn7_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 4:**
If a cardboard box (rectangular prism) with a top, a bottom, and 4 sides is \( W \) ft wide, \( L \) ft long, and \( H \) ft tall, then what is the surface area of this box? Find a formula for the surface area in terms of \( W \), \( L \), and \( H \). Explain clearly why your formula is valid.
**Explanation:**
To find the surface area of a rectangular prism (a cardboard box in this case), you need to calculate the area of all six faces and then sum them up.
A rectangular prism has three pairs of identical opposing faces:
1. The top and bottom faces.
2. The front and back faces.
3. The left and right side faces.
- The top and bottom faces each have an area of \( L \times W \).
- The front and back faces each have an area of \( L \times H \).
- The left and right side faces each have an area of \( W \times H \).
Thus, the total surface area \( A \) is calculated as follows:
\[ A = 2(L \times W) + 2(L \times H) + 2(W \times H) \]
This simplifies to:
\[ A = 2LW + 2LH + 2WH \]
This formula is valid because it correctly sums the areas of all six faces of the rectangular prism. Each dimension combination (\( LW \), \( LH \), and \( WH \)) appears twice in the process, corresponding to their pair of identical faces. This ensures that all surface areas of the box are included in the total calculation.
Therefore, the surface area \( A \) of the rectangular box in terms of \( W \), \( L \), and \( H \) is given by:
\[ A = 2(LW + LH + WH) \]
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