A cylinder of height 4 cm and base radius 3 cm is melted to form a sphere. The radius of sphere is:
3 cm
Cylinder Volume Calculation
The volume of a cylinder is given by the formula:
$ V_{cylinder} = \pi r^2 h $
Given:
- Cylinder base radius ($r$) = 3 cm
- Cylinder height ($h$) = 4 cm
Substituting the values:
$ V_{cylinder} = \pi \times (3 \text{ cm})^2 \times (4 \text{ cm}) $
$ V_{cylinder} = \pi \times 9 \text{ cm}^2 \times 4 \text{ cm} $
$ V_{cylinder} = 36\pi \text{ cm}^3 $
Sphere Volume Calculation
The volume of a sphere is given by the formula:
$ V_{sphere} = \frac{4}{3} \pi R^3 $
Where $R$ is the radius of the sphere.
Equating Volumes
When the cylinder is melted to form a sphere, the volume remains constant. Therefore:
$ V_{cylinder} = V_{sphere} $
$ 36\pi \text{ cm}^3 = \frac{4}{3} \pi R^3 $
Solving for Sphere Radius
To find the radius $R$ of the sphere, we solve the equation:
- Cancel $\pi$ from both sides:
$ 36 = \frac{4}{3} R^3 $
- Isolate $R^3$:
$ R^3 = 36 \times \frac{3}{4} $
$ R^3 = 9 \times 3 $
$ R^3 = 27 $
- Find the cube root:
$ R = \sqrt[3]{27 \text{ cm}^3} $
$ R = 3 \text{ cm} $
The radius of the sphere is 3 cm.
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