A regular pyramid has a square base. The height of the pyramid is 22 cm and side of its base is 14 cm. Volume of pyramid is equal to the volume of a sphere. What is the radius (in cm) of the sphere?

Options

A. `root(3)(49)`
B. 7
C. 14
D. `root(3)(98)`

Answer

To find the radius of the sphere whose volume is equal to the volume of a pyramid with a square base, we can follow these steps:

### Step 1: Calculate the Volume of the Pyramid
The formula for the volume \( V \) of a pyramid is given by:

\[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \]

For a pyramid with a square base, the area of the base can be calculated as:

\[ \text{Base Area} = \text{side}^2 = 14 \, \text{cm} \times 14 \, \text{cm} = 196 \, \text{cm}^2 \]

Now, substituting the base area and height into the volume formula:

\[ V = \frac{1}{3} \times 196 \, \text{cm}^2 \times 22 \, \text{cm} \]

Calculating this gives:

\[ V = \frac{1}{3} \times 4312 \, \text{cm}^3 = 1437.33 \, \text{cm}^3 \]

### Step 2: Set the Volume of the Sphere Equal to the Volume of the Pyramid
The volume \( V \) of a sphere is given by:

\[ V = \frac{4}{3} \pi r^3 \]

Setting the volume of the sphere equal to the volume of the pyramid:

\[ \frac{4}{3} \pi r^3 = 1437.33 \]

### Step 3: Solve for the Radius \( r \)
To find \( r^3 \), we first isolate it:

\[ r^3 = \frac{1437.33 \times 3}{4 \pi} \]

Calculating \( \frac{1437.33 \times 3}{4 \pi} \):

\[ r^3 = \frac{4312}{4 \pi} \]

Using \( \pi \approx 3.14 \):

\[ r^3 = \frac{4312}{12.56} \approx 343 \]

### Step 4: Take the Cube Root to Find \( r \)
Now, we take the cube root of both sides:

\[ r = \sqrt[3]{343} = 7 \, \text{cm} \]

### Final Answer
The radius of the sphere is:

\[ \boxed{7 \, \text{cm}} \]