A right cone has a base with diameter 10 units. The volume of the cone is 100𝜋 cubic units.
What is the length of a segment drawn from the apex to the edge of the circular base?
Make sure your finding the length of a segment
To find the length of the segment from the apex to the edge of the circular base in a cone, we first need to find the radius of the circular base.
The volume of a cone is given by the formula V = (1/3)πr^2h, where r is the radius of the base and h is the height of the cone.
Given that the volume of the cone is 100π cubic units, we have:
100π = (1/3)πr^2h
Since the diameter of the base is 10 units, the radius is half of that, which is 5 units. Let's substitute this into the formula:
100π = (1/3)π(5^2)h
100π = (1/3)π(25)h
100π = 25πh
h = 4 units
Now, we have the height of the cone as 4 units. To find the length of the segment from the apex to the edge of the circular base (the slant height), we can use the Pythagorean theorem:
Slant height = √(r^2 + h^2)
Slant height = √(5^2 + 4^2)
Slant height = √(25 + 16)
Slant height = √41 units
Therefore, the length of the segment from the apex to the edge of the circular base is √41 units.