Question
A sphere and a cylinder have the same radius and height. The volume of the cylinder is 54 meters cubed. Amie found the volume of the sphere.
A sphere with height h and radius r. A cylinder with height h and radius r.
Her work is shown below.
V = two-thirds + 54. V = two-thirds + StartFraction 162 Over 3 EndFraction. V = StartFraction 164 Over 3 EndFraction meter cubed.
What is Amie’s error?
Amie should have multiplied 54 by Two-thirds.
Amie should have multiplied 54 by Four-thirds pi.
Amie should have multiplied 54 by Four-thirds.
Amie should have multiplied 54 by Two-thirds pi.
A sphere with height h and radius r. A cylinder with height h and radius r.
Her work is shown below.
V = two-thirds + 54. V = two-thirds + StartFraction 162 Over 3 EndFraction. V = StartFraction 164 Over 3 EndFraction meter cubed.
What is Amie’s error?
Amie should have multiplied 54 by Two-thirds.
Amie should have multiplied 54 by Four-thirds pi.
Amie should have multiplied 54 by Four-thirds.
Amie should have multiplied 54 by Two-thirds pi.
Answers
GPT-4o mini
To determine Amie's error, we need to look at how the volume of a sphere and the volume of a cylinder are calculated.
1. The **volume of the cylinder** is given by the formula:
\[
V_{\text{cylinder}} = \pi r^2 h
\]
In this case, we know the volume of the cylinder is 54 cubic meters.
2. The **volume of the sphere** is given by the formula:
\[
V_{\text{sphere}} = \frac{4}{3} \pi r^3
\]
Since the cylinder and sphere have the same radius \( r \) and the same height \( h \), we need to relate the height with the radius.
We can find the relationship using the cylinder's volume:
\[
V_{\text{cylinder}} = \pi r^2 h = 54
\]
From this, we can express the product \( \pi r^2 h \) as \( 54 \).
Now, to find the volume of the sphere, we note that we need to express \( r \) in terms of \( h \) and \( V_{\text{cylinder}} \).
Given the relationship, we cannot directly use the cylinder's volume to find the sphere's volume by simply adding or multiplying by fractions. To find the volume of the sphere, Amie should have used the volume formula for the sphere.
### Analysis of Amie's steps:
Amie's work shows:
\[
V = \text{two-thirds} + 54
\]
This is not logically consistent since the volume of the sphere is not obtained by simply adding \( 54 \) to a fraction.
### Correct Calculation:
To find the **volume of the sphere** based on the relationship:
- Amie should have started with the equation specifically for calculating the sphere's volume.
The correct volume of the sphere can be represented as:
\[
V_{\text{sphere}} = \frac{4}{3} \pi r^3
\]
However, she neglected to use the actual formulas correctly, leading her to mistakenly add 54 instead of calculating \( \frac{4}{3} \pi \left(\frac{54}{\pi h}\right)^{3/2} \) using the correct radius.
Thus, from the options provided:
- **Amie should have multiplied 54 by Two-thirds pi.**
The correct reasoning would imply that to find the sphere's volume, the calculation involves using the relationship between the radius and height properly, thus leading her to compute it incorrectly.
1. The **volume of the cylinder** is given by the formula:
\[
V_{\text{cylinder}} = \pi r^2 h
\]
In this case, we know the volume of the cylinder is 54 cubic meters.
2. The **volume of the sphere** is given by the formula:
\[
V_{\text{sphere}} = \frac{4}{3} \pi r^3
\]
Since the cylinder and sphere have the same radius \( r \) and the same height \( h \), we need to relate the height with the radius.
We can find the relationship using the cylinder's volume:
\[
V_{\text{cylinder}} = \pi r^2 h = 54
\]
From this, we can express the product \( \pi r^2 h \) as \( 54 \).
Now, to find the volume of the sphere, we note that we need to express \( r \) in terms of \( h \) and \( V_{\text{cylinder}} \).
Given the relationship, we cannot directly use the cylinder's volume to find the sphere's volume by simply adding or multiplying by fractions. To find the volume of the sphere, Amie should have used the volume formula for the sphere.
### Analysis of Amie's steps:
Amie's work shows:
\[
V = \text{two-thirds} + 54
\]
This is not logically consistent since the volume of the sphere is not obtained by simply adding \( 54 \) to a fraction.
### Correct Calculation:
To find the **volume of the sphere** based on the relationship:
- Amie should have started with the equation specifically for calculating the sphere's volume.
The correct volume of the sphere can be represented as:
\[
V_{\text{sphere}} = \frac{4}{3} \pi r^3
\]
However, she neglected to use the actual formulas correctly, leading her to mistakenly add 54 instead of calculating \( \frac{4}{3} \pi \left(\frac{54}{\pi h}\right)^{3/2} \) using the correct radius.
Thus, from the options provided:
- **Amie should have multiplied 54 by Two-thirds pi.**
The correct reasoning would imply that to find the sphere's volume, the calculation involves using the relationship between the radius and height properly, thus leading her to compute it incorrectly.