SOLUTION: A cylinder is inscribed in a cone with height 10 and a base of radius 5, as shown below on the link. Find the approximate values of r and h for which the volume of the cylinder is

Algebra ->  Volume -> SOLUTION: A cylinder is inscribed in a cone with height 10 and a base of radius 5, as shown below on the link. Find the approximate values of r and h for which the volume of the cylinder is       Log On


   

Question 912071: A cylinder is inscribed in a cone with height 10 and a base of radius 5, as shown below on the link. Find the approximate values of r and h for which the volume of the cylinder is a maximum. Then give the approximate maximum volume.
http://postimg.org/image/tewn41qu7/
Thank you!!
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
There is a relationship between r and h.
When r=5, h=0
When r=0, h=10.
So then using the point-slope form of a line,
h-10=%28%2810-0%29%2F%280-5%29%29%28r-0%29
h-10=-2r
h=10-2r
So the volume of the cylinder is,
V=pi%2Ar%5E2%2Ah
V=pi%2Ar%5E2%2810-2r%29
To maximize the volume, take the derivative of the volume and set it equal to zero (use the chain rule).
dV%2Fdt=pi%2A%28r%5E2%28-2%29%2B%2810-2r%29%282r%29%29
dV%2Fdt=pi%2A%28-2r%5E2%2B%2820r-4r%5E2%29%29
dV%2Fdt=pi%2A%2820r-6r%5E2%29
dV%2Fdt=2pi%2Ar%2810-3r%29
So then,
10-3r=0
-3r=-10
r=10%2F3
Then,
h=10-2%2810%2F3%29
h=10-20%2F3
h=30%2F3-20%2F3
h=10%2F3
.
.
.
V%5Bmax%5D=pi%2A%2810%2F3%29%5E2%2A%2810%2F3%29
V%5Bmax%5D=%281000%2F27%29pi