Cylinder with maximum volume

Hello Godfather Quote:

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**Godfather** A piece of wire 60 inches long is cut into six sections, two of one length and four of another length. Each of the two sections having the same length is bent into the form of a circle and the two circles are then joined by the four remaining sections to make a frame for a model of a right circular cylinder.

(a). Find the lengths of the sections which will make the cylinder of maximum volume.

(b). Justify (a).

Suppose the length of one of the two equal lengths is $\displaystyle x$ inches. Then:

- Find, in terms of $\displaystyle x$, the radius of the circle into which one of these wires can be bent, using circumference of circle = $\displaystyle 2\pi r$
- Find the length of one of the four equal lengths, by subtracting $\displaystyle 2x$ from $\displaystyle 60$ and dividing the result by $\displaystyle 4$. This gives the height of the cylinder in terms of $\displaystyle x$.
- Use the formula for the volume of a cylinder to write down an expression for the volume, $\displaystyle V$, in terms of $\displaystyle x$.
- Use $\displaystyle \frac{dV}{dx}=0$ to find the value of $\displaystyle x$ which makes the value of $\displaystyle V$ a maximum or minimum.
- Check that this value gives a maximum.

Hope you can do it from here.

Grandad