What radius would equal 24(pi) times this radius?

For my algebra class, this is the only question im stuck on this section:

Grain silos can be described as a hemisphere sitting atop a cylinder. The interior volume V of the silo can be modeled by V = 2/3(pi)r^{3} + (pi)r^{2}h Where h=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 24(pi) times this radius?

That's the problem, word-for-word verbatim. 2/3 is a fraction and pi is the symbol for (pi). I'm lost (Speechless) How do I use the Latex thing so I can type it out exactly as it appears?

Any help would be greatly appreciated.

Re: What radius would equal 24(pi) times this radius?

Quote:

Originally Posted by

**Rtedmonston** For my algebra class, this is the only question im stuck on this section:

Grain silos can be described as a hemisphere sitting atop a cylinder. The interior volume V of the silo can be modeled by V = 2/3(pi)r^{3} + (pi)r^{2}h Where h=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 24(pi) times this radius?

That's the problem, word-for-word verbatim. 2/3 is a fraction and pi is the symbol for (pi). I'm lost (Speechless) How do I use the Latex thing so I can type it out exactly as it appears?

Any help would be greatly appreciated.

You only have to "translate" the text into an equation:

$\displaystyle \frac23 \pi r^3+\pi \cdot r^2 \cdot h = 24 \pi \cdot r$

Replace h by 6:

$\displaystyle \frac23 \pi r^3+6 \pi \cdot r^2 = 24 \pi \cdot r$

Collect all terms on the LHS and factor out $\displaystyle \pi r$ :

$\displaystyle \pi r \left( \frac23 r^2+6 \cdot r - 24 \right)=0$

A product of 2 factors equals zero if at least one factor equals zero. By the 1st factor you'll get r = 0 which is a very, very slim cylinder, nearly invisible (Rofl)

With the 2nd factor you have to solve a quadratic equation in r:

$\displaystyle \frac23 r^2+6 \cdot r - 24 = 0$

You'll get 2 solutions for r. The negative value isn't very plausible here.