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)
--
3
2) find r and h
3) what is the maximum volume
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)
--
3
2) find r and h
3) what is the maximum volume
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.