A metallic sphere of diameter 20 cm is recast into a right circular cone of base radius 10 cm. What is the bigger height of the cone ?

Options

A. 4 cm
B. 40 cm
C. 60 cm
D. 120 cm

Answer

To solve the problem, we need to find the height of a right circular cone that is formed by recasting a metallic sphere. We will follow these steps:

### Step 1: Calculate the volume of the sphere
The formula for the volume \( V \) of a sphere is given by:
\[ V = \frac{4}{3} \pi r^3 \]
where \( r \) is the radius of the sphere.

Given the diameter of the sphere is 20 cm, the radius \( r \) is:
\[ r = \frac{20}{2} = 10 \text{ cm} \]

Now, substituting the radius into the volume formula:
\[ V = \frac{4}{3} \pi (10)^3 = \frac{4}{3} \pi (1000) = \frac{4000}{3} \pi \text{ cm}^3 \]

### Step 2: Calculate the volume of the cone
The formula for the volume \( V \) of a cone is given by:
\[ V = \frac{1}{3} \pi r^2 h \]
where \( r \) is the radius of the base and \( h \) is the height of the cone.

Given the base radius of the cone is 10 cm, we can substitute this into the volume formula:
\[ V = \frac{1}{3} \pi (10)^2 h = \frac{1}{3} \pi (100) h = \frac{100}{3} \pi h \text{ cm}^3 \]

### Step 3: Set the volumes equal
Since the sphere is recast into the cone, their volumes are equal:
\[ \frac{4000}{3} \pi = \frac{100}{3} \pi h \]

### Step 4: Solve for height \( h \)
We can cancel \( \pi \) from both sides:
\[ \frac{4000}{3} = \frac{100}{3} h \]

Now, multiply both sides by 3 to eliminate the fraction:
\[ 4000 = 100h \]

Now, divide both sides by 100:
\[ h = \frac{4000}{100} = 40 \text{ cm} \]

### Conclusion
The height of the cone is \( 40 \) cm.

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