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Question 1 (Multiple Choice Worth 2 points)

A cone has a volume of [tex]$50 \pi \, \text{in}^3$[/tex] and a diameter of 10 in. Wilson states that a cylinder with the same height and diameter has the same volume. Which statement explains whether or not Wilson is correct?

A. A cylinder in which [tex]$h = 2$[/tex] and [tex]$d = 10$[/tex] has a volume of [tex]$50 \pi \, \text{in}^3$[/tex], therefore, Wilson is correct.

B. A cylinder in which [tex]$h = 6$[/tex] and [tex]$d = 10$[/tex] has a volume of [tex]$50 \pi \, \text{in}^3$[/tex], therefore, Wilson is correct.

C. A cylinder in which [tex]$h = 2$[/tex] and [tex]$d = 10$[/tex] has a volume of [tex]$150 \pi \, \text{in}^3$[/tex], therefore, Wilson is incorrect.

D. A cylinder in which [tex]$h = 6$[/tex] and [tex]$d = 10$[/tex] has a volume of [tex]$150 \pi \, \text{in}^3$[/tex], therefore, Wilson is incorrect.

Sagot :

GinnyAnswer
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.
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