The radius of the base of a cone and the radius of a sphere are the same, each being 8 cm. Given that the volumes of these two solids are also the same, calculate the slant height of the cone.

Answer

To solve the problem step by step, we will follow these steps:

### Step 1: Identify the given values
- The radius of the base of the cone (r) = 8 cm
- The radius of the sphere (r) = 8 cm
- The volumes of the cone and the sphere are equal.

### Step 2: Write the formulas for the volumes
- Volume of a cone (V_cone) = \(\frac{1}{3} \pi r^2 h\)
- Volume of a sphere (V_sphere) = \(\frac{4}{3} \pi r^3\)

### Step 3: Set the volumes equal to each other
Since the volumes are equal:
\[ \frac{1}{3} \pi r^2 h = \frac{4}{3} \pi r^3 \]

### Step 4: Cancel common terms
We can cancel \(\pi\) and \(\frac{1}{3}\) from both sides:
\[ r^2 h = 4 r^3 \]

### Step 5: Solve for the height (h) of the cone
Now, divide both sides by \(r^2\) (where \(r = 8\) cm):
\[ h = 4r \]
Substituting \(r = 8\) cm:
\[ h = 4 \times 8 = 32 \text{ cm} \]

### Step 6: Calculate the slant height (l) of the cone
The formula for the slant height (l) of a cone is given by:
\[ l = \sqrt{h^2 + r^2} \]
Substituting \(h = 32\) cm and \(r = 8\) cm:
\[ l = \sqrt{32^2 + 8^2} \]
Calculating the squares:
\[ l = \sqrt{1024 + 64} = \sqrt{1088} \]

### Step 7: Simplify the square root
Calculating \(\sqrt{1088}\):
\[ l \approx 32.98 \text{ cm} \]

### Final Answer
Hence, the slant height of the cone is approximately \(32.98\) cm.

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