Find the cone transcribing a prism

mercredi 30 octobre 2013

1. The problem statement, all variables and given/known data

Find the minimum value of the volume of a cone that is transcribing a four-sided prism with a=42cm and h=8cm

2. Relevant equations

[itex]V=\frac{r^{2}H\pi}{3}[/itex]

[itex]

\frac{H}{r}=\frac{h}{r-\frac{a}{2}}

[/itex]

3. The attempt at a solution

From the equation above it follows that

[itex]H=\frac{2hr}{2r-a}[/itex]

Inserting this into V I get

[itex]V=\frac{2hr^{3}\pi}{6r-3a}[/itex]

Taking the derivative of this I get

[itex]\frac{dr}{dV}=\frac{2r^{2}\pi (4r-3a)}{(2r-a)^{2}}[/itex]

Setting this 0 and solving for r I get

[itex]r=\frac{3a}{4}[/itex]

Since a is 42cm I get 31.5cm as the answer

Setting this into the H equation I get:

[itex]H=3h[/itex]

Since h is 8cm, I get 24cm as the answer

Now, my solution textbook tells me that I got the answer for H right, but it tells me that r should be

[itex]r=\frac{63\sqrt{2}}{2}[/itex]

What did I get wrong? Where does the square root come from? :/