A spheres has a volume of `36pi`. What is the surface area of the sphere? (The surface area of a sphere is given by the formula `A=4pir^(2)`)

Options

A. `3pi`
B. `9pi`
C. `27pi`
D. `36pi`

Answer

To find the surface area of a sphere given its volume, we can follow these steps:

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

### Step 2: Set the volume equal to the given value.
We know from the problem that the volume of the sphere is \( 36\pi \). Therefore, we can set up the equation:
\[ \frac{4}{3} \pi r^3 = 36\pi \]

### Step 3: Simplify the equation.
To simplify, we can divide both sides of the equation by \( \pi \):
\[ \frac{4}{3} r^3 = 36 \]

### Step 4: Multiply both sides by \( \frac{3}{4} \) to isolate \( r^3 \).
\[ r^3 = 36 \times \frac{3}{4} \]
Calculating the right side:
\[ r^3 = 27 \]

### Step 5: Take the cube root of both sides to find \( r \).
\[ r = \sqrt[3]{27} = 3 \]

### Step 6: Write down the formula for the surface area of a sphere.
The surface area \( A \) of a sphere is given by the formula:
\[ A = 4\pi r^2 \]

### Step 7: Substitute the value of \( r \) into the surface area formula.
Now we substitute \( r = 3 \) into the surface area formula:
\[ A = 4\pi (3^2) \]
Calculating \( 3^2 \):
\[ A = 4\pi (9) \]
\[ A = 36\pi \]

### Conclusion:
The surface area of the sphere is \( 36\pi \).

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