The volume of a right circular cone is `74 cm^(3)` and area of its base is `18.5 cm^(2)`. Find its height.

Answer

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

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

The area \( A \) of the base of the cone (which is a circle) is given by:
\[ A = \pi r^2 \]

### Step 2: Substitute the given values
We know:
- Volume \( V = 74 \, \text{cm}^3 \)
- Area \( A = 18.5 \, \text{cm}^2 \)

From the area formula, we can express \( r^2 \):
\[ \pi r^2 = 18.5 \]
Thus, we can find \( r^2 \):
\[ r^2 = \frac{18.5}{\pi} \]

### Step 3: Substitute \( r^2 \) into the volume formula
Now, we substitute \( r^2 \) into the volume formula:
\[ 74 = \frac{1}{3} \pi \left(\frac{18.5}{\pi}\right) h \]

### Step 4: Simplify the equation
The \( \pi \) in the numerator and denominator cancels out:
\[ 74 = \frac{1}{3} \cdot 18.5 \cdot h \]

### Step 5: Multiply both sides by 3
To eliminate the fraction, multiply both sides by 3:
\[ 3 \cdot 74 = 18.5 \cdot h \]
\[ 222 = 18.5 \cdot h \]

### Step 6: Solve for \( h \)
Now, divide both sides by 18.5 to find \( h \):
\[ h = \frac{222}{18.5} \]

### Step 7: Calculate \( h \)
Calculating the division:
\[ h = 12 \, \text{cm} \]

Thus, the height of the cone is \( 12 \, \text{cm} \).

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