The volume of a right circular cone is 100 `pi` `cm^3` and its height is 12 cm. Find its slant height.

Options

A. 13 cm
B. 16 cm
C. 9cm
D. 26 cm

Answer

To find the slant height of a right circular cone given its volume and height, we can follow these steps:

### Step 1: Write down the formula for the volume of a cone.
The volume \( V \) of a cone is given by the formula:
\[ V = \frac{1}{3} \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height of the cone.

### Step 2: Substitute the known values into the volume formula.
We know that the volume \( V = 100 \pi \, \text{cm}^3 \) and the height \( h = 12 \, \text{cm} \). Substituting these values into the formula gives:
\[ 100 \pi = \frac{1}{3} \pi r^2 \cdot 12 \]

### Step 3: Simplify the equation.
First, we can cancel \( \pi \) from both sides:
\[ 100 = \frac{1}{3} r^2 \cdot 12 \]
Next, multiply both sides by 3 to eliminate the fraction:
\[ 300 = 12 r^2 \]

### Step 4: Solve for \( r^2 \).
Now, divide both sides by 12:
\[ r^2 = \frac{300}{12} = 25 \]

### Step 5: Find the radius \( r \).
Taking the square root of both sides gives:
\[ r = \sqrt{25} = 5 \, \text{cm} \]

### Step 6: Use the radius to find the slant height \( L \).
The slant height \( L \) of a cone can be found using the formula:
\[ L = \sqrt{r^2 + h^2} \]
Substituting the values of \( r \) and \( h \):
\[ L = \sqrt{5^2 + 12^2} \]

### Step 7: Calculate \( L \).
Calculating the squares:
\[ L = \sqrt{25 + 144} = \sqrt{169} \]
Taking the square root gives:
\[ L = 13 \, \text{cm} \]

### Final Answer:
The slant height of the cone is \( 13 \, \text{cm} \).

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