# Thread: Volum of a cylinder in a cone (driving me insane!)

1. ## Volum of a cylinder in a cone (driving me insane!)

a cylinder of radius r and height h is inscribed within a cone with bas radius 6 and height 20

1) show that the volume of the cylinder is given by

V= 10.(pi).r^2.(6-r)
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3

2) find r and h

3) what is the maximum volume

2. First, what is the relationship between the height of the cylinder and the radius of the cylinder?

If you make a sketch, you'll see that for a certain radius r, the cone would have a height of l = (r/6)*20, using similar triangles.

From here, the height of the cylinder will become h = 20 - (20r)/6.

Then, the formula for the volume of a cylinder is

$\displaystyle V = \pi r^2 h$

Using the equation you got, you now get:

$\displaystyle V = \pi r^2 (20 - \frac{20r}{6})$

Simplifying;

$\displaystyle V = \frac{10}{3}\pi r^2 (6-r)$

For the second part, are you sure that's the question? Because r and h would vary, depending on the volume. If the question was in fact, "Find the value of r and h, provided that the volume of the cylinder is maximised." there there would be a solution.

3. thanks. it says to find h and r when they are maximum values so i think i shouldve posted this in calculus

4. Yes, you need to find the derivative of V, and solve for V' at V' = 0.

You will get the radius in this situation, and you can then find the height using r.

5. ok thanks