If a cone with a slant height equal to the diameter of the base is inscribed in a sphere with a radius of 10, what is the volume of the cone?
A. $375\pi$
B. $300\pi$
C. $250\pi$
D. $200\pi$
E. $160\pi$
Looks like the diameter and the slant heights of the cone would form an equilateral triangle. I used such a relationship but then I can't seem to get the right answer.
When viewed along a diameter as line of sight we have an equilateral triangle.
$$ Vol= \frac13\cdot \pi (5 \sqrt3)^2\cdot 15 = 375 \pi$$
Option A.
It's a simple question.
Hints: Use radius of sphere as the distance between the equilateral triangle's centroid and the vertices of equilateral triangle.Giving you radius $$r=10 \times \cos 30^o$$ $$r=5\sqrt{3}$$ $$diameter=slant (L)=10\sqrt{3} units$$
Now, height of cone(h) can be solved via: $L^2-r^2=h^2$ $$h=15 units$$
Now apply $$Volume=\frac{1}{3}\pi r^2 h$$ To get $375 \pi $ units as answer
