A sphere has the same curved surface area as a cone of height 40 cm and radius 30 cm. The radius of the sphere is

Options

A. `5sqrt(5) cm`
B. `5sqrt(3) cm`
C. `5sqrt(15) cm`
D. `5sqrt(10) cm`

Answer

To find the radius of the sphere that has the same curved surface area as the given cone, we will follow these steps:

### Step 1: Calculate the slant height of the cone
The slant height (L) of the cone can be calculated using the Pythagorean theorem. The formula is:

\[ L = \sqrt{h^2 + r^2} \]

Where:
- \( h \) is the height of the cone (40 cm)
- \( r \) is the radius of the cone (30 cm)

Calculating:

\[ L = \sqrt{40^2 + 30^2} = \sqrt{1600 + 900} = \sqrt{2500} = 50 \text{ cm} \]

### Step 2: Calculate the curved surface area of the cone
The curved surface area (CSA) of the cone is given by the formula:

\[ \text{CSA}_{\text{cone}} = \pi r L \]

Substituting the values:

\[ \text{CSA}_{\text{cone}} = \pi \times 30 \times 50 = 1500\pi \text{ cm}^2 \]

### Step 3: Set the curved surface area of the sphere equal to that of the cone
The curved surface area of a sphere is given by:

\[ \text{CSA}_{\text{sphere}} = 4\pi R^2 \]

Where \( R \) is the radius of the sphere. Since the curved surface areas are equal, we set them equal to each other:

\[ 4\pi R^2 = 1500\pi \]

### Step 4: Simplify the equation
We can cancel \( \pi \) from both sides:

\[ 4R^2 = 1500 \]

Dividing both sides by 4:

\[ R^2 = \frac{1500}{4} = 375 \]

### Step 5: Solve for R
Taking the square root of both sides:

\[ R = \sqrt{375} = \sqrt{25 \times 15} = 5\sqrt{15} \text{ cm} \]

Thus, the radius of the sphere is \( 5\sqrt{15} \) cm.

### Summary of the Solution
The radius of the sphere is \( 5\sqrt{15} \) cm.

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