To find out which set of dimensions for the cone gives it the same volume as the cylinder, we need to use the formulas for the volume of a cylinder and a cone.
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Volume of a Cylinder: \[ V_{\text{cylinder}} = \pi r_{\text{cylinder}}^2 h_{\text{cylinder}} \]
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Volume of a Cone: \[ V_{\text{cone}} = \frac{1}{3} \pi r_{\text{cone}}^2 h_{\text{cone}} \]
Given that the diameter of the cylinder is 10, the radius \( r_{\text{cylinder}} \) is \( \frac{10}{2} = 5 \).
Let’s consider \( h_{\text{cylinder}} \) to determine the volume of the cylinder. We can compute a general expression for volume in relation to the height of the cylinder.
- Volume of the Cylinder (assuming height = 4): \[ V_{\text{cylinder}} = \pi (5)^2 (4) = 100\pi \]
Now, we need to check which of the cone dimensions provided would result in a cone volume of \( 100\pi \).
Options for the cone:
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Radius = 5, Height = 12: \[ V_{\text{cone}} = \frac{1}{3} \pi (5^2)(12) = \frac{1}{3} \pi (25)(12) = \frac{300}{3} \pi = 100\pi \]
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Radius = 5, Height = 4: \[ V_{\text{cone}} = \frac{1}{3} \pi (5^2)(4) = \frac{1}{3} \pi (25)(4) = \frac{100}{3} \pi \neq 100\pi \]
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Radius = 15, Height = 4: \[ V_{\text{cone}} = \frac{1}{3} \pi (15^2)(4) = \frac{1}{3} \pi (225)(4) = \frac{900}{3} \pi = 300\pi \neq 100\pi \]
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Radius = 15, Height = 12: \[ V_{\text{cone}} = \frac{1}{3} \pi (15^2)(12) = \frac{1}{3} \pi (225)(12) = \frac{2700}{3} \pi = 900\pi \neq 100\pi \]
Conclusion:
The only dimensions for the cone that equal the volume of the cylinder when the cylinder has a diameter of 10 and height of 4 is:
- Radius = 5, Height = 12.