Question : The radius of a sphere and that of the base of a cylinder are equal. The ratio of the radius of the base of the cylinder and the height of the cylinder is 3 : 4. What is the ratio of the volume of the sphere to that of the cylinder?
Option 1: 27 : 64
Option 2: 1 : 2
Option 3: 1 : 1
Option 4: 9 : 16
Correct Answer: 1 : 1
Solution :
Radius of the sphere = Radius of the cylinder (let it be $r$)
The ratio of the radius to the height of the cylinder = 3 : 4
Volume of Sphere = $\frac{4}{3} \pi r^3$
Volume of Cylinder = $\pi r^2 h$
Let the height of the cylinder be $4x$ (since the ratio of radius to height is 3 : 4)
Then, the radius of the cylinder, $r = 3x$
Therefore, Volume of Sphere = $\frac{4}{3} \pi (3\text{x})^3$ = $36\pi \text{x}^3$
And, Volume of Cylinder = $\pi (3\text{x})^2 \times 4\text{x} = 36\pi \text{x}^3$
Ratio of the volume of sphere to that of the cylinder = $\frac{36\pi \text{x}^3}{36\pi \text{x}^3}$ = 1 : 1
Hence, the ratio of the volume of the sphere to that of the cylinder is 1 : 1.
Hence, the correct answer is 1 : 1.
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