### Determining the Volume of a Storage Shed **Problem 15: A Volume Problem** The front and back of a storage shed are shaped like isosceles right triangles with two sides of length 10 ft, as shown on the left of **Figure 13.37**. The storage shed is 40 ft long. Determine the volume of the storage shed. --- **Solution Process:** 1. **Using the Pythagorean Theorem:** - Qing first uses the Pythagorean theorem to determine that the length of the unknown side of the triangle is \(\sqrt{200}\) ft long. - **Figure 13.37 Front View:** - A right isosceles triangle is shown with each of the equal sides measuring 10 ft. 2. **Calculating the Height of the Triangle:** - Qing uses the Pythagorean theorem again to calculate the height of the triangle \(h\) if the base is the side of length \(\sqrt{200}\). Qing carries this out as follows: \[ \left(\frac{\sqrt{200}}{2}\right)^2 + h^2 = 10^2 \] \[ h^2 = 100 - \frac{200}{4} = 100 - 50 = 50 \] \[ h = \sqrt{50} \] 3. **Determining the Floor Area:** - Next, Qing determines that the floor area of the storage shed is: \[ 40 \cdot \sqrt{200} \text{ square feet} \] 4. **Calculating the Volume:** - Finally, Qing calculates that the volume of the storage shed is given by the floor area of the shed times the height of the triangle: \[ 40 \cdot \sqrt{200} \cdot \sqrt{50} \] --- **Discussion Questions:** a. **Is Qing’s method of calculation correct? Discuss Qing’s work. Which parts (if any) are correct; which parts (if any) are incorrect?** b. **Solve the volume problem in a different way than Qing did, explaining your reasoning.** --- Figure 13.37: This figure shows a front view of the storage shed's triangular face. The triangle is an isosceles right triangle
### Determining the Volume of a Storage Shed **Problem 15: A Volume Problem** The front and back of a storage shed are shaped like isosceles right triangles with two sides of length 10 ft, as shown on the left of **Figure 13.37**. The storage shed is 40 ft long. Determine the volume of the storage shed. --- **Solution Process:** 1. **Using the Pythagorean Theorem:** - Qing first uses the Pythagorean theorem to determine that the length of the unknown side of the triangle is \(\sqrt{200}\) ft long. - **Figure 13.37 Front View:** - A right isosceles triangle is shown with each of the equal sides measuring 10 ft. 2. **Calculating the Height of the Triangle:** - Qing uses the Pythagorean theorem again to calculate the height of the triangle \(h\) if the base is the side of length \(\sqrt{200}\). Qing carries this out as follows: \[ \left(\frac{\sqrt{200}}{2}\right)^2 + h^2 = 10^2 \] \[ h^2 = 100 - \frac{200}{4} = 100 - 50 = 50 \] \[ h = \sqrt{50} \] 3. **Determining the Floor Area:** - Next, Qing determines that the floor area of the storage shed is: \[ 40 \cdot \sqrt{200} \text{ square feet} \] 4. **Calculating the Volume:** - Finally, Qing calculates that the volume of the storage shed is given by the floor area of the shed times the height of the triangle: \[ 40 \cdot \sqrt{200} \cdot \sqrt{50} \] --- **Discussion Questions:** a. **Is Qing’s method of calculation correct? Discuss Qing’s work. Which parts (if any) are correct; which parts (if any) are incorrect?** b. **Solve the volume problem in a different way than Qing did, explaining your reasoning.** --- Figure 13.37: This figure shows a front view of the storage shed's triangular face. The triangle is an isosceles right triangle
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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![### Determining the Volume of a Storage Shed
**Problem 15: A Volume Problem**
The front and back of a storage shed are shaped like isosceles right triangles with two sides of length 10 ft, as shown on the left of **Figure 13.37**. The storage shed is 40 ft long. Determine the volume of the storage shed.
---
**Solution Process:**
1. **Using the Pythagorean Theorem:**
- Qing first uses the Pythagorean theorem to determine that the length of the unknown side of the triangle is \(\sqrt{200}\) ft long.
- **Figure 13.37 Front View:**
- A right isosceles triangle is shown with each of the equal sides measuring 10 ft.
2. **Calculating the Height of the Triangle:**
- Qing uses the Pythagorean theorem again to calculate the height of the triangle \(h\) if the base is the side of length \(\sqrt{200}\). Qing carries this out as follows:
\[
\left(\frac{\sqrt{200}}{2}\right)^2 + h^2 = 10^2
\]
\[
h^2 = 100 - \frac{200}{4} = 100 - 50 = 50
\]
\[
h = \sqrt{50}
\]
3. **Determining the Floor Area:**
- Next, Qing determines that the floor area of the storage shed is:
\[
40 \cdot \sqrt{200} \text{ square feet}
\]
4. **Calculating the Volume:**
- Finally, Qing calculates that the volume of the storage shed is given by the floor area of the shed times the height of the triangle:
\[
40 \cdot \sqrt{200} \cdot \sqrt{50}
\]
---
**Discussion Questions:**
a. **Is Qing’s method of calculation correct? Discuss Qing’s work. Which parts (if any) are correct; which parts (if any) are incorrect?**
b. **Solve the volume problem in a different way than Qing did, explaining your reasoning.**
---
Figure 13.37: This figure shows a front view of the storage shed's triangular face. The triangle is an isosceles right triangle](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7c60e9dc-9541-469d-8098-00c54f69ea68%2F01437950-cf4d-4473-8960-d07cbc733f3d%2Feohhhcm_processed.png&w=3840&q=75)
Transcribed Image Text:### Determining the Volume of a Storage Shed
**Problem 15: A Volume Problem**
The front and back of a storage shed are shaped like isosceles right triangles with two sides of length 10 ft, as shown on the left of **Figure 13.37**. The storage shed is 40 ft long. Determine the volume of the storage shed.
---
**Solution Process:**
1. **Using the Pythagorean Theorem:**
- Qing first uses the Pythagorean theorem to determine that the length of the unknown side of the triangle is \(\sqrt{200}\) ft long.
- **Figure 13.37 Front View:**
- A right isosceles triangle is shown with each of the equal sides measuring 10 ft.
2. **Calculating the Height of the Triangle:**
- Qing uses the Pythagorean theorem again to calculate the height of the triangle \(h\) if the base is the side of length \(\sqrt{200}\). Qing carries this out as follows:
\[
\left(\frac{\sqrt{200}}{2}\right)^2 + h^2 = 10^2
\]
\[
h^2 = 100 - \frac{200}{4} = 100 - 50 = 50
\]
\[
h = \sqrt{50}
\]
3. **Determining the Floor Area:**
- Next, Qing determines that the floor area of the storage shed is:
\[
40 \cdot \sqrt{200} \text{ square feet}
\]
4. **Calculating the Volume:**
- Finally, Qing calculates that the volume of the storage shed is given by the floor area of the shed times the height of the triangle:
\[
40 \cdot \sqrt{200} \cdot \sqrt{50}
\]
---
**Discussion Questions:**
a. **Is Qing’s method of calculation correct? Discuss Qing’s work. Which parts (if any) are correct; which parts (if any) are incorrect?**
b. **Solve the volume problem in a different way than Qing did, explaining your reasoning.**
---
Figure 13.37: This figure shows a front view of the storage shed's triangular face. The triangle is an isosceles right triangle
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