What is the height of a cylinder that has the same volume and radius as a sphere of diameter 12 cm ?

Options

A. a.7 cm
B. b.10 cm
C. c.9 cm
D. d.8 cm

Answer

To find the height of a cylinder that has the same volume and radius as a sphere with a diameter of 12 cm, we can follow these steps:

### Step 1: Find the radius of the sphere
The diameter of the sphere is given as 12 cm. The radius (r) is half of the diameter.
\[ r = \frac{diameter}{2} = \frac{12 \, \text{cm}}{2} = 6 \, \text{cm} \]

### Step 2: Write the formula for the volume of the sphere
The volume (V) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]

### Step 3: Substitute the radius into the volume formula
Now, we substitute the radius we found into the volume formula:
\[ V = \frac{4}{3} \pi (6)^3 \]

### Step 4: Calculate \(6^3\)
Calculate \(6^3\):
\[ 6^3 = 216 \]
So, the volume becomes:
\[ V = \frac{4}{3} \pi (216) = \frac{864}{3} \pi = 288 \pi \, \text{cm}^3 \]

### Step 5: Write the formula for the volume of the cylinder
The volume (V) of a cylinder is given by the formula:
\[ V = \pi r^2 h \]
Where \(h\) is the height of the cylinder.

### Step 6: Set the volumes equal to each other
Since the volume of the sphere is equal to the volume of the cylinder:
\[ 288 \pi = \pi (6^2) h \]

### Step 7: Simplify the equation
We can cancel \(\pi\) from both sides:
\[ 288 = 36h \]

### Step 8: Solve for \(h\)
Now, we solve for \(h\):
\[ h = \frac{288}{36} = 8 \, \text{cm} \]

### Conclusion
The height of the cylinder is \(8 \, \text{cm}\).

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