Step 1: Formula for the Volume of the Cone
The formula for the volume \( V \) of a cone is: \[ V = \frac{1}{3} \pi r^2 h \] where: - \( V \) is the volume, - \( r \) is the radius of the base, - \( h \) is the height of the cone. Given that: - \( V = 48\pi \, \text{cm}^3 \), - \( h = 9 \, \text{cm} \), we can substitute the known values into the formula: \[ 48\pi = \frac{1}{3} \pi r^2 \times 9 \]
Step 2: Solving for the Radius
First, divide both sides of the equation by \( \pi \): \[ 48 = \frac{1}{3} \times 9 \times r^2 \] Simplify: \[ 48 = 3r^2 \] Now, divide both sides by 3: \[ r^2 = 16 \] Taking the square root of both sides: \[ r = 4 \, \text{cm} \]
Step 3: Finding the Diameter
The diameter \( d \) is twice the radius: \[ d = 2r = 2 \times 4 = 8 \, \text{cm} \]
The diameter of the base of the cone is \( 8 \, \text{cm} \).