This problem involves finding the diameter of a cone given its volume and height. We'll use the formula for the volume of a cone to solve for the radius, and then double the radius to find the diameter....
The Formula for Cone Volume
The volume (V) of a cone is calculated using the following formula:
V = (1/3)πr²h
Where:
- r is the radius of the cone's base
- h is the height of the cone
- π is a mathematical constant approximately equal to 3.14159
Solving for the Radius
We know the volume (V) is 20π cubic meters and the height (h) is 10 meters. Let's substitute these values into the formula and solve for the radius (r):
20π = (1/3)πr² * 10
Simplifying the equation:
20π = (10/3)πr²
Dividing both sides by (10/3)π:
20π / ((10/3)π) = r²
This simplifies to:
6 = r²
Taking the square root of both sides:
√6 = r
Therefore, the radius (r) of the cone is approximately √6 meters.
Calculating the Diameter
The diameter (d) of a circle is twice its radius. So, the diameter of the cone's base is:
d = 2r = 2 * √6 ≈ 4.9 meters
Answer
Therefore, the diameter of the cone is approximately **4.9 meters** when rounded to the nearest tenth.