A wooden block is in the shape of the right circular cylinder shown above. If the volume of the wooden block is 64π cubic centimeters, what is the diameter of the base of the cylin... A wooden block is in the shape of the right circular cylinder shown above. If the volume of the wooden block is 64π cubic centimeters, what is the diameter of the base of the cylinder, in centimeters?
Understand the Problem
The question is asking to find the diameter of the base of a cylinder, given its height and volume. We need to use the formula for the volume of a cylinder, which is V = πr²h, where r is the radius and h is the height. The volume is provided, so we will set up the equation to find r, and then calculate the diameter from r.
Answer
The diameter of the base of the cylinder is $4$ centimeters.
Answer for screen readers
The diameter of the base of the cylinder is ( 4 ) centimeters.
Steps to Solve
- Volume Formula for a Cylinder The formula for the volume of a cylinder is given by:
$$ V = \pi r^2 h $$
Where:
- ( V ) is the volume
- ( r ) is the radius
- ( h ) is the height
- Substitute the Known Values We know the volume ( V = 64\pi ) cubic centimeters and the height ( h = 16 ) centimeters. Substitute these values into the volume formula:
$$ 64\pi = \pi r^2 \cdot 16 $$
- Simplify the Equation Divide both sides by ( \pi ):
$$ 64 = r^2 \cdot 16 $$
- Solve for ( r^2 ) Divide both sides by 16 to isolate ( r^2 ):
$$ r^2 = \frac{64}{16} $$
- Calculate the Radius Now calculate ( \frac{64}{16} ):
$$ r^2 = 4 $$
Taking the square root to find ( r ):
$$ r = \sqrt{4} = 2 $$
- Calculate the Diameter The diameter ( d ) is twice the radius:
$$ d = 2r = 2 \cdot 2 = 4 $$
The diameter of the base of the cylinder is ( 4 ) centimeters.
More Information
The volume of a cylinder can be visualized as the area of the circular base multiplied by its height. In this case, using the properties of the cylinder and the given dimensions led us to determine the radius and, subsequently, the diameter.
Tips
- Ignoring Units: Always ensure to check the units for consistency, especially when calculating volume.
- Misapplying the Formula: Ensure that the correct formula for volume is used.
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