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. option 1 2 3 or 4
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. option 1 2 3 or 4
Answers
GPT-4o mini
To find the volume of a sphere, the formula is:
\[
V = \frac{4}{3} \pi r^3
\]
Given that the cylinder is said to have a volume of 8 cubic meters, this must be referring to the volume of the cylinder, which is given by:
\[
V = \pi r^2 h
\]
Since the sphere and the cylinder have the same radius \( r \) and height \( h \), and in this case, we can relate the volume of the cylinder to that of the sphere by calculating their volumes with the same radius \( r \).
Yolanda seems to have mistakenly taken the volume of the cylinder (8 cubic meters) as if it represented \( r \) cubed directly. Instead, she should be using the radius itself (not the volume of the cylinder) in the sphere volume formula.
From the volume of the cylinder, we are not directly given the radius or height but the volume itself. Hence, she has misunderstood the substitution. The cube of the radius should not be taken to be the volume of the cylinder.
Now, examining her work:
1. "V = four-thirds (8) cubed." is clearly incorrect because \( 8 \) should not be cubed; \( r \) should be derived from the cylinder's volume.
2. She is incorrectly assuming \( 8 \) is the radius.
Let's analyze the provided options:
1. **Yolanda should have found the volume by multiplying 8 by Two-thirds.** - Incorrect usage.
2. **Yolanda should have found the volume by multiplying 8 by Four-thirds.** - This is incorrect; she needs to find the radius first.
3. **Yolanda should have found the volume with the formula V = two-thirds pi (8) cubed.** - Incorrect formula and values.
4. **Yolanda should have found the volume with the formula V = two-thirds (8) cubed.** - Also incorrect because the formula does not include \(\pi\), and it misuses the 8.
None of the options directly addresses the mistake in using the cylinder's volume \( V \) in an inappropriate way regarding the radius's use in calculating sphere volume.
However, looking at the choices closest to the right direction:
- **Yolanda should have found the volume with the formula V = two-thirds (8) cubed.** could describe her mistaken attempt to use the cylinder's volume instead of deriving the correct radius needed to find the sphere's volume correctly.
Therefore, option **4** resembles her error closest.
\[
V = \frac{4}{3} \pi r^3
\]
Given that the cylinder is said to have a volume of 8 cubic meters, this must be referring to the volume of the cylinder, which is given by:
\[
V = \pi r^2 h
\]
Since the sphere and the cylinder have the same radius \( r \) and height \( h \), and in this case, we can relate the volume of the cylinder to that of the sphere by calculating their volumes with the same radius \( r \).
Yolanda seems to have mistakenly taken the volume of the cylinder (8 cubic meters) as if it represented \( r \) cubed directly. Instead, she should be using the radius itself (not the volume of the cylinder) in the sphere volume formula.
From the volume of the cylinder, we are not directly given the radius or height but the volume itself. Hence, she has misunderstood the substitution. The cube of the radius should not be taken to be the volume of the cylinder.
Now, examining her work:
1. "V = four-thirds (8) cubed." is clearly incorrect because \( 8 \) should not be cubed; \( r \) should be derived from the cylinder's volume.
2. She is incorrectly assuming \( 8 \) is the radius.
Let's analyze the provided options:
1. **Yolanda should have found the volume by multiplying 8 by Two-thirds.** - Incorrect usage.
2. **Yolanda should have found the volume by multiplying 8 by Four-thirds.** - This is incorrect; she needs to find the radius first.
3. **Yolanda should have found the volume with the formula V = two-thirds pi (8) cubed.** - Incorrect formula and values.
4. **Yolanda should have found the volume with the formula V = two-thirds (8) cubed.** - Also incorrect because the formula does not include \(\pi\), and it misuses the 8.
None of the options directly addresses the mistake in using the cylinder's volume \( V \) in an inappropriate way regarding the radius's use in calculating sphere volume.
However, looking at the choices closest to the right direction:
- **Yolanda should have found the volume with the formula V = two-thirds (8) cubed.** could describe her mistaken attempt to use the cylinder's volume instead of deriving the correct radius needed to find the sphere's volume correctly.
Therefore, option **4** resembles her error closest.