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
Related questions
Question
![**Title: Calculating the Surface Area of a Cylindrical Can**
---
**Objective:**
Learn how to calculate the surface area of a cylindrical object using the example of a soup can.
---
**Problem Statement:**
**Figure 1:**
In the image, we have a cylindrical can of Progresso Garden Vegetable Soup. The following measurements are provided:
- Diameter of the can: 2 inches
- Height of the can: 4.5 inches
---
**Steps to Calculate the Surface Area:**
1. **Identify the Dimensions:**
- Radius (r) = Diameter/2 = 2 inches / 2 = 1 inch
- Height (h) = 4.5 inches
2. **Surface Area Formula:**
To find the surface area of a cylinder, we use the formula:
\[
A = 2\pi r(h + r)
\]
where \( A \) is the surface area, \( r \) is the radius, and \( h \) is the height.
3. **Calculate:**
Substituting the given values:
\[
A = 2\pi \times 1 \times (4.5 + 1)
\]
Simplifying further:
\[
A = 2\pi \times 1 \times 5.5
\]
Therefore:
\[
A = 11\pi \approx 34.56 \, \text{square inches}
\]
---
**Conclusion:**
Using the given dimensions, the surface area of this can of soup is approximately 34.56 square inches. This calculation includes both the lateral surface area and the area of the two bases (top and bottom) of the cylinder.
---
**Note:**
The image depicts a can with key measurements indicated by dotted lines to help visualize the dimensions necessary for the calculations.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6723c038-ec47-4bfd-80a5-b112ac1b61f2%2F371054c8-ca77-420f-abc2-876e9c2c4d46%2F1j0kvnq_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Calculating the Surface Area of a Cylindrical Can**
---
**Objective:**
Learn how to calculate the surface area of a cylindrical object using the example of a soup can.
---
**Problem Statement:**
**Figure 1:**
In the image, we have a cylindrical can of Progresso Garden Vegetable Soup. The following measurements are provided:
- Diameter of the can: 2 inches
- Height of the can: 4.5 inches
---
**Steps to Calculate the Surface Area:**
1. **Identify the Dimensions:**
- Radius (r) = Diameter/2 = 2 inches / 2 = 1 inch
- Height (h) = 4.5 inches
2. **Surface Area Formula:**
To find the surface area of a cylinder, we use the formula:
\[
A = 2\pi r(h + r)
\]
where \( A \) is the surface area, \( r \) is the radius, and \( h \) is the height.
3. **Calculate:**
Substituting the given values:
\[
A = 2\pi \times 1 \times (4.5 + 1)
\]
Simplifying further:
\[
A = 2\pi \times 1 \times 5.5
\]
Therefore:
\[
A = 11\pi \approx 34.56 \, \text{square inches}
\]
---
**Conclusion:**
Using the given dimensions, the surface area of this can of soup is approximately 34.56 square inches. This calculation includes both the lateral surface area and the area of the two bases (top and bottom) of the cylinder.
---
**Note:**
The image depicts a can with key measurements indicated by dotted lines to help visualize the dimensions necessary for the calculations.
![**Educational Website Content:**
---
### Calculating the Volume of a Basketball
**Problem:**
**Find the VOLUME of my basketball with a radius of 4.75 inches.**
![Basketball Image](https://example.com/image) *(Image showing a basketball.)
To solve for the volume of a basketball, we need to use the formula for the volume of a sphere. This formula is:
\[ V = \frac{4}{3} \pi r^3 \]
Where:
- \( V \) is the volume.
- \( r \) is the radius of the sphere.
- \( \pi \) (Pi) is approximately 3.14159.
Given that the radius \( r = 4.75 \) inches, we can substitute this value into the formula:
\[ V = \frac{4}{3} \times 3.14159 \times (4.75)^3 \]
Calculating the above expressions step-by-step:
1. First calculate \( (4.75)^3 \):
\[ (4.75)^3 = 4.75 \times 4.75 \times 4.75 = 107.171875 \]
2. Then, multiply by \( \pi \):
\[ \pi \times 107.171875 \approx 3.14159 \times 107.171875 = 336.24408 \]
3. Finally, multiply by \( \frac{4}{3} \):
\[ V = \frac{4}{3} \times 336.24408 \approx 1.33333 \times 336.24408 = 448.32544 \]
So, the volume of the basketball is approximately \( 448.33 \) cubic inches.
---
**Graph/Diagram Explanation:**
- The image above shows a standard basketball.
- The black lines on the basketball represent the typical pattern found on standard basketballs.
- For the purpose of this calculation, imagine slicing the basketball in half to see that it is a hollow sphere, which inherently helps understand where the 4.75 inches radius is measured from - it is the distance from the center of the sphere to any point on its surface.
By following the above steps, students can learn how to practically apply the volume formula for spheres to real-life objects like a basketball.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6723c038-ec47-4bfd-80a5-b112ac1b61f2%2F371054c8-ca77-420f-abc2-876e9c2c4d46%2Fnmxmk7o_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Educational Website Content:**
---
### Calculating the Volume of a Basketball
**Problem:**
**Find the VOLUME of my basketball with a radius of 4.75 inches.**
![Basketball Image](https://example.com/image) *(Image showing a basketball.)
To solve for the volume of a basketball, we need to use the formula for the volume of a sphere. This formula is:
\[ V = \frac{4}{3} \pi r^3 \]
Where:
- \( V \) is the volume.
- \( r \) is the radius of the sphere.
- \( \pi \) (Pi) is approximately 3.14159.
Given that the radius \( r = 4.75 \) inches, we can substitute this value into the formula:
\[ V = \frac{4}{3} \times 3.14159 \times (4.75)^3 \]
Calculating the above expressions step-by-step:
1. First calculate \( (4.75)^3 \):
\[ (4.75)^3 = 4.75 \times 4.75 \times 4.75 = 107.171875 \]
2. Then, multiply by \( \pi \):
\[ \pi \times 107.171875 \approx 3.14159 \times 107.171875 = 336.24408 \]
3. Finally, multiply by \( \frac{4}{3} \):
\[ V = \frac{4}{3} \times 336.24408 \approx 1.33333 \times 336.24408 = 448.32544 \]
So, the volume of the basketball is approximately \( 448.33 \) cubic inches.
---
**Graph/Diagram Explanation:**
- The image above shows a standard basketball.
- The black lines on the basketball represent the typical pattern found on standard basketballs.
- For the purpose of this calculation, imagine slicing the basketball in half to see that it is a hollow sphere, which inherently helps understand where the 4.75 inches radius is measured from - it is the distance from the center of the sphere to any point on its surface.
By following the above steps, students can learn how to practically apply the volume formula for spheres to real-life objects like a basketball.
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