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Sagot :
Let's solve the problem step-by-step to determine if Wilson's claim about a cylinder's volume with given dimensions is correct.
### Given Data:
1. Cylinder diameter, [tex]\( d = 10 \)[/tex] inches.
2. Volume of a cone, [tex]\( V_{\text{cone}} = 50\pi \)[/tex] cubic inches.
We are checking the volumes of two cylinders with different heights:
1. Cylinder height, [tex]\( h = 2 \)[/tex] inches.
2. Cylinder height, [tex]\( h = 6 \)[/tex] inches.
### Step-by-Step Calculation:
1. Calculate the radius of the cylinder:
Since the diameter [tex]\( d = 10 \)[/tex] inches,
[tex]\[ \text{Radius, } r = \frac{d}{2} = \frac{10}{2} = 5 \text{ inches} \][/tex]
2. Volume of a cylinder is given by the formula:
[tex]\[ V = \pi r^2 h \][/tex]
3. Calculate the volume of the cylinder with height [tex]\( h = 2 \)[/tex] inches:
[tex]\[ V_1 = \pi (5)^2 (2) = \pi \cdot 25 \cdot 2 = 50\pi \text{ cubic inches} \][/tex]
4. Calculate the volume of the cylinder with height [tex]\( h = 6 \)[/tex] inches:
[tex]\[ V_2 = \pi (5)^2 (6) = \pi \cdot 25 \cdot 6 = 150\pi \text{ cubic inches} \][/tex]
### Conclusion:
- For a cylinder with [tex]\( d = 10 \)[/tex] inches and [tex]\( h = 2 \)[/tex] inches, the volume is [tex]\( 50\pi \)[/tex] cubic inches. Hence, Wilson is correct in stating that this volume matches the volume of the cone, [tex]\( 50\pi \)[/tex] cubic inches.
- For a cylinder with [tex]\( d = 10 \)[/tex] inches and [tex]\( h = 6 \)[/tex] inches, the volume is [tex]\( 150\pi \)[/tex] cubic inches, which does not match the volume of the cone.
### Answer Choices Evaluation:
- Option 1: A cylinder in which [tex]\( h = 2 \)[/tex] and [tex]\( d = 10 \)[/tex] has a volume of [tex]\( 50 \pi \)[/tex] in [tex]\( ^3 \)[/tex]; therefore, Wilson is correct. (True)
- Option 2: A cylinder in which [tex]\( h = 6 \)[/tex] and [tex]\( d = 10 \)[/tex] has a volume of [tex]\( 50 \pi \)[/tex] in [tex]\( ^3 \)[/tex]; therefore, Wilson is correct. (False)
- Option 3: A cylinder in which [tex]\( h = 2 \)[/tex] and [tex]\( d = 10 \)[/tex] has a volume of [tex]\( 150 \pi \)[/tex] in [tex]\( ^3 \)[/tex]; therefore, Wilson is incorrect. (False)
- Option 4: A cylinder in which [tex]\( h = 6 \)[/tex] and [tex]\( d = 10 \)[/tex] has a volume of [tex]\( 150 \pi \)[/tex] in [tex]\( ^3 \)[/tex]; therefore, Wilson is incorrect. (True)
The answers to the multiple-choice options are Option 1 and Option 4.
### Given Data:
1. Cylinder diameter, [tex]\( d = 10 \)[/tex] inches.
2. Volume of a cone, [tex]\( V_{\text{cone}} = 50\pi \)[/tex] cubic inches.
We are checking the volumes of two cylinders with different heights:
1. Cylinder height, [tex]\( h = 2 \)[/tex] inches.
2. Cylinder height, [tex]\( h = 6 \)[/tex] inches.
### Step-by-Step Calculation:
1. Calculate the radius of the cylinder:
Since the diameter [tex]\( d = 10 \)[/tex] inches,
[tex]\[ \text{Radius, } r = \frac{d}{2} = \frac{10}{2} = 5 \text{ inches} \][/tex]
2. Volume of a cylinder is given by the formula:
[tex]\[ V = \pi r^2 h \][/tex]
3. Calculate the volume of the cylinder with height [tex]\( h = 2 \)[/tex] inches:
[tex]\[ V_1 = \pi (5)^2 (2) = \pi \cdot 25 \cdot 2 = 50\pi \text{ cubic inches} \][/tex]
4. Calculate the volume of the cylinder with height [tex]\( h = 6 \)[/tex] inches:
[tex]\[ V_2 = \pi (5)^2 (6) = \pi \cdot 25 \cdot 6 = 150\pi \text{ cubic inches} \][/tex]
### Conclusion:
- For a cylinder with [tex]\( d = 10 \)[/tex] inches and [tex]\( h = 2 \)[/tex] inches, the volume is [tex]\( 50\pi \)[/tex] cubic inches. Hence, Wilson is correct in stating that this volume matches the volume of the cone, [tex]\( 50\pi \)[/tex] cubic inches.
- For a cylinder with [tex]\( d = 10 \)[/tex] inches and [tex]\( h = 6 \)[/tex] inches, the volume is [tex]\( 150\pi \)[/tex] cubic inches, which does not match the volume of the cone.
### Answer Choices Evaluation:
- Option 1: A cylinder in which [tex]\( h = 2 \)[/tex] and [tex]\( d = 10 \)[/tex] has a volume of [tex]\( 50 \pi \)[/tex] in [tex]\( ^3 \)[/tex]; therefore, Wilson is correct. (True)
- Option 2: A cylinder in which [tex]\( h = 6 \)[/tex] and [tex]\( d = 10 \)[/tex] has a volume of [tex]\( 50 \pi \)[/tex] in [tex]\( ^3 \)[/tex]; therefore, Wilson is correct. (False)
- Option 3: A cylinder in which [tex]\( h = 2 \)[/tex] and [tex]\( d = 10 \)[/tex] has a volume of [tex]\( 150 \pi \)[/tex] in [tex]\( ^3 \)[/tex]; therefore, Wilson is incorrect. (False)
- Option 4: A cylinder in which [tex]\( h = 6 \)[/tex] and [tex]\( d = 10 \)[/tex] has a volume of [tex]\( 150 \pi \)[/tex] in [tex]\( ^3 \)[/tex]; therefore, Wilson is incorrect. (True)
The answers to the multiple-choice options are Option 1 and Option 4.
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