Each of the measure of the radius of base of a cone and that of a sphere is 8 cm. Also, the volume of these two solids are equal. The slant height of the cone is

Options

A. `8sqrt(17) cm`
B. `4sqrt(17) cm`
C. `34sqrt(2) cm`
D. 34 cm

Answer

To find the slant height of the cone given that the radius of the base of the cone and the sphere is 8 cm and their volumes are equal, we can follow these steps:

### Step 1: Calculate the volume of the sphere.
The formula for the volume of a sphere is:
\[ V_{\text{sphere}} = \frac{4}{3} \pi r^3 \]
Given that the radius \( r = 8 \) cm, we can substitute this value into the formula:
\[ V_{\text{sphere}} = \frac{4}{3} \pi (8)^3 = \frac{4}{3} \pi (512) = \frac{2048}{3} \pi \text{ cm}^3 \]

### Step 2: Set the volume of the cone equal to the volume of the sphere.
The formula for the volume of a cone is:
\[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \]
Since the radius of the cone is also 8 cm, we can write:
\[ V_{\text{cone}} = \frac{1}{3} \pi (8)^2 h = \frac{1}{3} \pi (64) h = \frac{64}{3} \pi h \text{ cm}^3 \]
Setting the volumes equal gives us:
\[ \frac{64}{3} \pi h = \frac{2048}{3} \pi \]

### Step 3: Solve for height \( h \).
We can cancel \( \frac{\pi}{3} \) from both sides:
\[ 64h = 2048 \]
Now, divide both sides by 64:
\[ h = \frac{2048}{64} = 32 \text{ cm} \]

### Step 4: Calculate the slant height of the cone.
The slant height \( l \) of a cone can be calculated using the Pythagorean theorem:
\[ l = \sqrt{r^2 + h^2} \]
Substituting the known values:
\[ l = \sqrt{(8)^2 + (32)^2} = \sqrt{64 + 1024} = \sqrt{1088} \]

### Step 5: Simplify \( \sqrt{1088} \).
We can factor 1088:
\[ 1088 = 64 \times 17 \]
Thus,
\[ l = \sqrt{64 \times 17} = \sqrt{64} \times \sqrt{17} = 8\sqrt{17} \text{ cm} \]

### Final Answer:
The slant height of the cone is \( 8\sqrt{17} \) cm.

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