Answer :
Let's break it down step-by-step:
1. Given Information:
- Volume of the cone, \(V_{\text{cone}} = 50\pi \) cubic inches.
- Diameter of the cone, \( d = 10 \) inches.
- Radius of the cone (and hence the cylinder, since the diameter is the same), \( r = \frac{10}{2} = 5 \) inches.
2. Volume Formula for a Cone:
The volume \( V_{\text{cone}} \) of a cone is given by:
[tex]\[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \][/tex]
- We know \( V_{\text{cone}} = 50\pi \) cubic inches and \( r = 5 \) inches.
- Let's solve for \( h \) (height of the cone):
[tex]\[ 50\pi = \frac{1}{3} \pi (5)^2 h \][/tex]
[tex]\[ 50 = \frac{1}{3} (25) h \][/tex]
[tex]\[ 50 = \frac{25}{3} h \][/tex]
[tex]\[ 50 \cdot 3 = 25 h \][/tex]
[tex]\[ 150 = 25 h \][/tex]
[tex]\[ h = \frac{150}{25} = 6 \text{ inches} \][/tex]
- So the height of the cone is \( 6 \) inches.
3. Volume Formula for a Cylinder:
The volume \( V_{\text{cylinder}} \) of a cylinder is given by:
[tex]\[ V_{\text{cylinder}} = \pi r^2 h \][/tex]
- Let's analyze the given options based on this formula.
4. Evaluating the Choices:
- Option A: A cylinder with \( h = 2 \) inches and \( d = 10 \) inches.
[tex]\[ r = 5 \text{ inches} \][/tex]
[tex]\[ V_{\text{cylinder}} = \pi (5)^2 (2) = 50\pi \text{ cubic inches} \][/tex]
- This matches the volume of the cone. Therefore, the statement is Wilson is correct for this option.
- Option B: A cylinder with \( h = 6 \) inches and \( d = 10 \) inches.
[tex]\[ r = 5 \text{ inches} \][/tex]
[tex]\[ V_{\text{cylinder}} = \pi (5)^2 (6) = 150\pi \text{ cubic inches} \][/tex]
- This volume does not match the volume of the cone. Therefore, the statement is Wilson is incorrect for this option.
- Option C: This option conflicts with valid mathematical principles, as it isn't a controlled comparison based on cone characteristics.
- Statement is confusing and incorrect.
- Option D: Repeats option B's argument with correct math.
[tex]\[ V_{\text{cylinder}} = 150\pi \text{ cubic inches} \][/tex]
- Therefore, the statement is Wilson is incorrect for this option.
Based on the evaluation, the correct choice that matches the context of the problem is:
D. A cylinder in which \( h = 6 \) inches and \( d = 10 \) inches has a volume of \( 150\pi \) cubic inches, therefore, Wilson is incorrect.
Therefore, the correct answer is:
4.
1. Given Information:
- Volume of the cone, \(V_{\text{cone}} = 50\pi \) cubic inches.
- Diameter of the cone, \( d = 10 \) inches.
- Radius of the cone (and hence the cylinder, since the diameter is the same), \( r = \frac{10}{2} = 5 \) inches.
2. Volume Formula for a Cone:
The volume \( V_{\text{cone}} \) of a cone is given by:
[tex]\[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \][/tex]
- We know \( V_{\text{cone}} = 50\pi \) cubic inches and \( r = 5 \) inches.
- Let's solve for \( h \) (height of the cone):
[tex]\[ 50\pi = \frac{1}{3} \pi (5)^2 h \][/tex]
[tex]\[ 50 = \frac{1}{3} (25) h \][/tex]
[tex]\[ 50 = \frac{25}{3} h \][/tex]
[tex]\[ 50 \cdot 3 = 25 h \][/tex]
[tex]\[ 150 = 25 h \][/tex]
[tex]\[ h = \frac{150}{25} = 6 \text{ inches} \][/tex]
- So the height of the cone is \( 6 \) inches.
3. Volume Formula for a Cylinder:
The volume \( V_{\text{cylinder}} \) of a cylinder is given by:
[tex]\[ V_{\text{cylinder}} = \pi r^2 h \][/tex]
- Let's analyze the given options based on this formula.
4. Evaluating the Choices:
- Option A: A cylinder with \( h = 2 \) inches and \( d = 10 \) inches.
[tex]\[ r = 5 \text{ inches} \][/tex]
[tex]\[ V_{\text{cylinder}} = \pi (5)^2 (2) = 50\pi \text{ cubic inches} \][/tex]
- This matches the volume of the cone. Therefore, the statement is Wilson is correct for this option.
- Option B: A cylinder with \( h = 6 \) inches and \( d = 10 \) inches.
[tex]\[ r = 5 \text{ inches} \][/tex]
[tex]\[ V_{\text{cylinder}} = \pi (5)^2 (6) = 150\pi \text{ cubic inches} \][/tex]
- This volume does not match the volume of the cone. Therefore, the statement is Wilson is incorrect for this option.
- Option C: This option conflicts with valid mathematical principles, as it isn't a controlled comparison based on cone characteristics.
- Statement is confusing and incorrect.
- Option D: Repeats option B's argument with correct math.
[tex]\[ V_{\text{cylinder}} = 150\pi \text{ cubic inches} \][/tex]
- Therefore, the statement is Wilson is incorrect for this option.
Based on the evaluation, the correct choice that matches the context of the problem is:
D. A cylinder in which \( h = 6 \) inches and \( d = 10 \) inches has a volume of \( 150\pi \) cubic inches, therefore, Wilson is incorrect.
Therefore, the correct answer is:
4.
