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A right circular cone has a volume of 140 in^3. The height of the cone is the same length as the diameter of the base. Find the radius and height.

We know the formula for the Volume of a Right Circular Cone is given by
V=(1/3)*pi*r^2*h
V=140 in^3
The height of the cone = diameter of the base. The diameter = 2 times the radius, so h = 2r
The formula for Volume can now be written as
V=(1/3)*pi*r^2*(2r)
which simplifies to
V=(2/3)*pi*r^3
You plug in 140 in^3 for V and solve for r. Then you can plug the value you find for r into the equation h=2r

V = (1/3 π r^2 h , but h = 2r
3V = π r^2 (2r) = 2π r^3
420 = 2πr^3
r^3 = 210/π
r = (210/π)^(1/3) = 4.0584

r = 4.0584
h = 8.11683

check:
V = (1/3)π(4.0584)^2 (8.11683) = 139.9989.. , not bad

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