Question

The radius of a right cylinder is 15 inches and its height is 6 inches. What single amount can be added to either the radius or the height so that the volumes will increase the same amount?

Answer 1

we want

pi * (r+x)^2 h = pi r^2 (h+x)

hr^2 + 2rhx + hx^2 = hr^2 + xr^2

2rhx + hx^2 = xr^2

hx^2 + (2rh - r^2)x = 0
x(hx + 2rh - r^2) = 0

So, x=0 (add nothing)
or x = (r^2 - 2rh)/h = r/h * (r-2h)

Check:

h(r^2/h - r)^2 = r^2 (h + r^2/h - 2r)

r^4/h - 2r^3 + hr^2 = hr^2 + r^4/h - 2r^3