Hello there. To understand this geometry problem, we may need to do some drawings:
First, the cone and the cylinder:
We know that the base of the cone is equal to b and its height is equal to a.
Before using the last informations given by the question, we need to remember the properties of the right circular cylinder:
The lateral area of a cylinder is given by 2*pi* radius * height, since when we "unroll" the cylinder, we have something like in the following drawing:
For the cone, its lateral area is given by:
In this case, the cone geratrix can be expressed in terms of a and b.
If the base of the cone is b, then its radius is equal to b/2.
Thus, its geratrix is equal to g = sqrt(a² + (b/2)²) = sqrt(a² + b²/4).
Now, its lateral area will be equal to: pi * b/2 * sqrt(a² + b²/4)
Before solving for the radius of the cylinder, using that information "cone which surmounts the cylinder" is important to find the height of the cylinder.
Using triangle congruence properties, we have:
Cross multiplying, we get:
ab/2 + ar - h(b/2 + r) = ar
ab/2 = h(b/2 + r)
h = ab/(b + 2r)
Now, finally, we can solve for r using the lateral areas:
2pi*r*ab/(b+2r) = pi*b/2*sqrt(a² + b²/4)
Divide both sides by 2pi
r*ab/(b+2r) = b*sqrt(a² + b²/4)/4
Then, solving for r, we get:
r = -(b sqrt(4 a^2 + b^2))/(2 (sqrt(4 a^2 + b^2) - 4 a))
As in the drawing:
