SOLUTION: Find the volume of the largest cylinder that can be inscribed in a right circular cone of radius 3 inches and whose height is 10 inches.
Algebra.Com
Question 1122184: Find the volume of the largest cylinder that can be inscribed in a right circular cone of radius 3 inches and whose height is 10 inches.
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!
See diagram. The isosceles triangle is a representation of a cross-section of the given cone. The rectangle is an arbitrary cylinder inscribed in the cone. The cross-section of the cone is placed on a coordinate grid such that the center of the base of the cone is at the origin and the apex of the cone is at the point (0,10). That means the intersection of a generator of the cone and a radius of the base of the cone is at the point (3,0).
Using the two points, (0,10) and (3,0), and designating the coordinate axes as 
and )
, we can write an equation of the right-hand cone generator as \ =\ -\frac{10}{3}(r\ -\ 3))
. From the diagram, we can see that represents the radius of the cylinder and represents the height of the cylinder as a function of the radius. Note the domain of this function is the open interval (0,3) because the endpoints would represent zero volume cylinders.
The volume of a cylinder is given by the area of the base times the height. Using the information developed so far, we can write a function that represents the volume of the cylinder as a function of the radius of its base.
\ =\ \pi{r^2}\(-\frac{10}{3}\(r\ -\ 3\)\)\ =\ 10\pi\(-\frac{r^3}{3}\ +\ r^2\))
Then
)
Note 
is not in the domain of either the cone generator function or the cylinder volume function. Clearly, there is an extremum at 
. We could go through the process of checking the second derivative to ensure that this is a maximum, but intuitively it is pretty clear that this has to be a maximum since the volume near the endpoints of the function domain approaches zero and we have a substantial volume at and near
\ =\ 10\pi\(-\frac{2^3}{3}\ +\ 2^2\))
\ =\ 10\pi\(4\ -\ \frac{8}{3}\)\ =\ \frac{40\pi}{3}\text{ in^3})
John
My calculator said it, I believe it, that settles it
RELATED QUESTIONS
Find the volume of the largest cylinder with circular base that can be inscribed in a... (answered by ankor@dixie-net.com)
Find the dimensions of the right circular cylinder of greatest volume that can be... (answered by greenestamps)
find the volume of the largest cylinder with a circular base that can be inscribed in a... (answered by lwsshak3,jamesgunasekaran)
What is the volume of the largest right circular cylinder that can be inscribed in a cube (answered by Alan3354)
find the volume of the largest circular cone that can be cut out of a cube whose edge is... (answered by Gogonati)
Find the dimensions and volume of the right circular cylinder of maximum volume inscribed (answered by josgarithmetic)
The diameter of the base of a right circular cone is 10 in., and its altitude is 8 in.... (answered by greenestamps)
find the equation that represents the volume of a right circular cone whose radius is t... (answered by Fombitz)
17. Find the volume of the largest right circular cylinder of altitude 8 in. that can be... (answered by CPhill,greenestamps)