Question : What is the height of a cylinder that has the same volume and radius as a sphere of diameter 12 cm?
Option 1: 7 cm
Option 2: 10 cm
Option 3: 9 cm
Option 4: 8 cm
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Correct Answer: 8 cm
Solution :
The volume of a sphere = $\frac{4}{3}\pi r_{\text{s}}^3$
Where $r_{\text{s}}$ is the radius of the sphere.
The volume of a cylinder = $\pi r_{\text{c}}^2 h_{\text{c}}$
Where $r_{\text{c}}$ is the radius of the base of the cylinder and $h_{\text{c}}$ is the height of the cylinder.
Given that the sphere and the cylinder have the same volume and the same radius.
$⇒\frac{4}{3}\pi r_{\text{s}}^3 = \pi r_{\text{c}}^2 h_{\text{c}}$
Since $r_{\text{s}} = r_{\text{c}} = \frac{12}{2}$ = 6 cm
$⇒\frac{4}{3}\pi (6)^3 = \pi (6)^2 h_{\text{c}}$
$⇒h_{\text{c}} = \frac{4}{3} (6) = 8$
Hence, the correct answer is 8 cm.
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