Question
A sphere and a cylinder have the same radius and height. The volume of the cylinder is 8 meters cubed. Yolanda 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 = four-thirds (8) cubed. V = four-thirds (512). V = StartFraction 2,048 Over 3 EndFraction meters cubed.
What is Yolanda’s error?
Yolanda should have found the volume by multiplying 8 by Two-thirds.
Yolanda should have found the volume by multiplying 8 by Four-thirds.
Yolanda should have found the volume with the formula V = two-thirds pi (8) cubed.
Yolanda should have found the volume with the formula V = two-thirds (8) cubed.
A sphere with height h and radius r. A cylinder with height h and radius r.
Her work is shown below.
V = four-thirds (8) cubed. V = four-thirds (512). V = StartFraction 2,048 Over 3 EndFraction meters cubed.
What is Yolanda’s error?
Yolanda should have found the volume by multiplying 8 by Two-thirds.
Yolanda should have found the volume by multiplying 8 by Four-thirds.
Yolanda should have found the volume with the formula V = two-thirds pi (8) cubed.
Yolanda should have found the volume with the formula V = two-thirds (8) cubed.
Answers
GPT-4o mini
Let's break down Yolanda's work and identify the error.
1. **Volume of the Cylinder**: Yolanda states that the volume of the cylinder is 8 cubic meters. The formula for the volume of a cylinder is:
\[
V_{cylinder} = \pi r^2 h
\]
Since the cylinder's volume is given as 8 m³, we can conclude the volume is correct for the dimensions provided (same radius r and height h).
2. **Volume of the Sphere**: The formula for the volume of a sphere is:
\[
V_{sphere} = \frac{4}{3} \pi r^3
\]
There isn't information about the radius directly, but we will later see if Yolanda uses the correct dimensions based on the cylinder's volume.
3. **Yolanda's Calculation**: Yolanda's work shows:
\[
V = \frac{4}{3} (8)^3
\]
Here, she seemingly is treating the cylinder's volume as if it relates directly to the sphere's volume calculation using \(8\) directly for \(r\). She does \( (8)^3 = 512 \), and then correctly proceeds to find:
\[
V = \frac{4}{3} (512) = \frac{2048}{3} \text{ m}^3
\]
4. **Identifying the Error**: Yolanda made a fundamental mistake in interpreting the use of the radius \(r\) (which should have been derived from the dimensions of the cylinder).
The correct approach would relate the sphere's volume rather than directly using the volume of the cylinder as if it were the radius. The radius must be determined from the given volume.
Since \(V_{cylinder} = 8\) m³ can be used to find \(r\), we should instead solve:
\[
V_{sphere} = \frac{4}{3} \pi r^3
\]
with the established parameters. The sphere's radius \(r\) would be derived from the cylinder dimensions or setting the calculations based on the formulas rather than using the volume value itself for radius.
Thus, the correct answer regarding the errors in her work is that:
**Yolanda should have found the volume by multiplying 8 by Four-thirds.**
This reflects the relationship of directly finding the radius correctly and using the appropriate value instead of incorrectly cubing the already provided volume.
1. **Volume of the Cylinder**: Yolanda states that the volume of the cylinder is 8 cubic meters. The formula for the volume of a cylinder is:
\[
V_{cylinder} = \pi r^2 h
\]
Since the cylinder's volume is given as 8 m³, we can conclude the volume is correct for the dimensions provided (same radius r and height h).
2. **Volume of the Sphere**: The formula for the volume of a sphere is:
\[
V_{sphere} = \frac{4}{3} \pi r^3
\]
There isn't information about the radius directly, but we will later see if Yolanda uses the correct dimensions based on the cylinder's volume.
3. **Yolanda's Calculation**: Yolanda's work shows:
\[
V = \frac{4}{3} (8)^3
\]
Here, she seemingly is treating the cylinder's volume as if it relates directly to the sphere's volume calculation using \(8\) directly for \(r\). She does \( (8)^3 = 512 \), and then correctly proceeds to find:
\[
V = \frac{4}{3} (512) = \frac{2048}{3} \text{ m}^3
\]
4. **Identifying the Error**: Yolanda made a fundamental mistake in interpreting the use of the radius \(r\) (which should have been derived from the dimensions of the cylinder).
The correct approach would relate the sphere's volume rather than directly using the volume of the cylinder as if it were the radius. The radius must be determined from the given volume.
Since \(V_{cylinder} = 8\) m³ can be used to find \(r\), we should instead solve:
\[
V_{sphere} = \frac{4}{3} \pi r^3
\]
with the established parameters. The sphere's radius \(r\) would be derived from the cylinder dimensions or setting the calculations based on the formulas rather than using the volume value itself for radius.
Thus, the correct answer regarding the errors in her work is that:
**Yolanda should have found the volume by multiplying 8 by Four-thirds.**
This reflects the relationship of directly finding the radius correctly and using the appropriate value instead of incorrectly cubing the already provided volume.