Question

grain silos can be described as a hemisphere sitting atop a cylinder. The interior volume V of the silo can be modeled by V=2/3πr^3+πr^2h, where h is the height of a cylinder with radius r. For a cylinder 6m tall, what radius would give the silo a volume that is numerically equal to 24π times the radius?

Answer 1

this is always fun...
\[24\pi r = 2/3 \pi r^3 + \pi r^2 \times6\]
divide by pi
Simplifying \[24 r = 2/3 r^3 + 6r^2\]

\[36r = r^3 + 9r^2\]
divide by r
\[r^2 + 9r - 36 = 0\]

solve the quadratic... and remember only the the positive r as length can only be positive