Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
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![**Pyramid Volume Calculation Problem**
---
**Problem Statement:**
The pyramid shown below has a square base, a height of 7 units, and a volume of 84 cubic units.
[Diagram of a Pyramid]
- There is a pictorial representation of a pyramid with a square base.
- The height of the pyramid is marked as 7 units.
**Question:**
What is the length of the side of the base?
**Choices:**
1) 6
2) 12
3) 18
4) 36
---
**Explanation of the Diagram:**
- The pyramid has a square base, meaning all four sides of the base are equal in length.
- The height of the pyramid, represented by a vertical line from the base to the apex, measures 7 units.
---
**Solution:**
To find the side length of the base of the square pyramid, we use the formula for the volume of a pyramid:
\[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \]
Given:
- Volume (\( V \)) = 84 cubic units
- Height (\( h \)) = 7 units
- Let side length of the base = \( s \)
1. Write the formula for the base area of a square pyramid:
\[ \text{Base Area} = s^2 \]
2. Substitute the known values into the volume formula:
\[ 84 = \frac{1}{3} \times s^2 \times 7 \]
3. Simplify the equation:
\[ 84 = \frac{7}{3} \times s^2 \]
4. Multiply both sides by 3 to solve for \( s^2 \):
\[ 84 \times 3 = 7 \times s^2 \]
\[ 252 = 7 \times s^2 \]
5. Divide both sides by 7:
\[ s^2 = \frac{252}{7} \]
\[ s^2 = 36 \]
6. Take the square root of both sides to find \( s \):
\[ s = \sqrt{36} \]
\[ s = 6 \]
So, the length of the side of the base is 6 units.
Thus, the correct answer is:
**1)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3985eccf-67c8-430e-916c-80bb763431d2%2F863fcde3-c8a1-48da-a2cb-4cdcfab259b5%2Fc0mytc8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Pyramid Volume Calculation Problem**
---
**Problem Statement:**
The pyramid shown below has a square base, a height of 7 units, and a volume of 84 cubic units.
[Diagram of a Pyramid]
- There is a pictorial representation of a pyramid with a square base.
- The height of the pyramid is marked as 7 units.
**Question:**
What is the length of the side of the base?
**Choices:**
1) 6
2) 12
3) 18
4) 36
---
**Explanation of the Diagram:**
- The pyramid has a square base, meaning all four sides of the base are equal in length.
- The height of the pyramid, represented by a vertical line from the base to the apex, measures 7 units.
---
**Solution:**
To find the side length of the base of the square pyramid, we use the formula for the volume of a pyramid:
\[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \]
Given:
- Volume (\( V \)) = 84 cubic units
- Height (\( h \)) = 7 units
- Let side length of the base = \( s \)
1. Write the formula for the base area of a square pyramid:
\[ \text{Base Area} = s^2 \]
2. Substitute the known values into the volume formula:
\[ 84 = \frac{1}{3} \times s^2 \times 7 \]
3. Simplify the equation:
\[ 84 = \frac{7}{3} \times s^2 \]
4. Multiply both sides by 3 to solve for \( s^2 \):
\[ 84 \times 3 = 7 \times s^2 \]
\[ 252 = 7 \times s^2 \]
5. Divide both sides by 7:
\[ s^2 = \frac{252}{7} \]
\[ s^2 = 36 \]
6. Take the square root of both sides to find \( s \):
\[ s = \sqrt{36} \]
\[ s = 6 \]
So, the length of the side of the base is 6 units.
Thus, the correct answer is:
**1)
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