# Math - Calculus I

Optimization Problem:

Find the dimensions of the right circular cylinder of greatest volume inscribed in a right circular cone of radius 10" and height 24"

1. 👍 0
2. 👎 0
3. 👁 375
1. Draw a side view of the situation
(The cone will look like an isosceles triangle with a rectangle (the cylinder) sitting on its base and touching the sides)

let the radius of the cylinder be r and let the height of the cylinder be h
Look at the small right-angled triangle at the right
we can set up a ratio because of similar triangles
h/(10-r) = 24/10
10h = 240-24r
h = 24 - 12r/5

V = πr^2 h
= πr^2 (24 - 12r/5)
=24πr^2 - (12π/5) r^3

dV/dr = 48πr - (36π/5)r2
= 0 for a max/min

48πr = (36π/5)r^2
÷ by 12π
4r = 3π/5 r^2
divide by r, since r ≠ 0, it would give a minimum volume
4 = 3r/5
20 = 3r
r = 20/3
then h = 24 - (12/5)(20/3) = 8

the cylinder has a radius of 20/3 and a height of 8

check:
with those dimensions, V = 1117.01
let r = 19/3 (a little less) , then h = 44/5 , V = 1108.9 , a bit less
let r = 21/3 ( a little more), then h = 36/5 , V = 1108.35 (again, a little less)

1. 👍 0
2. 👎 0

## Similar Questions

1. ### Calculus

A solid is formed by adjoining two hemispheres to the ends of a right circular cylinder. An industrial tank of this shape must have a volume of 1600 cubic feet. The hemispherical ends cost twice as much per square foot of surface

2. ### Calculus :)

A rectangle is inscribed with its base on the x-axis and its upper corners on the parabola y=6–x^2. What are the dimensions of such a rectangle with the greatest possible area? Find Width=____ & Height=4 just need to find

3. ### Math

The table on the right shows the measured dimensions of a rectangular prism and the minimum and maximum possible dimensions based on the greatest possible error. What is the greatest possible percent error in finding the volume of

4. ### Math HELP

The table below shows the measured dimensions of a prism, and the maximum and minimum possible values based on the greatest possible error. Dimensions l w h Measured 9 5 3 Maximum 9.5 5.5 3.5 Minimum 8.5 4.5 2.5 a. Find the

1. ### Calculus

A cylinder is inscribed in a right circular cone of height 5.5 and radius (at the base) equal to 2 . A) What are the dimensions of such a cylinder which has maximum volume? B) What is the radius? C) What is the height?

2. ### Calculus

A solid is formed by adjoining two hemispheres to the ends of a right circular cylinder. The total volume of the solid is 2 cubic centimeters. Find the radius of the cylinder that produces the minimum surface area. (Round your

3. ### Calculus

An energy drink container in the shape of a right circular cylinder must have a volume of 12 fluid ounces (1 fluid ounce is approximately 1.80469 cubic inches). The cost per square inch of constructing the top and bottom is twice

4. ### geometry

A cylinder is inscribed in a cube. If the edge of the cube is 3" long, find the volume of the cylinder.

1. ### science

A circular disk of 10 cm radius is charged uniformly with a total charge of Q coul. Find the electric field intensity at a point 20 cm away from the disk, along its axis. (b) Consider a uniformly charged right circular cylindrical

2. ### MATH

A cylinder has a circular base with a diameter of 12 ft. The height of the cylinder is 4 ft. What is the volume of the cylinder rounded to the nearest whole number? Use 3.14 for pi. A. 452in2 B. 1,809ft2 C. 151ft2 D. 603ft2

3. ### mathe

Show that a right-circular cylinder of greatest volume that can be inscribed in a right-circular cone that has a volume that is 4/9 the volume of the cone.

4. ### Algebra

The volume of a right circular cylinder (think of a pop can) is jointly proportional to the square of the radius of the circular base and to the height. For example, when the height is 10.62 cm and the radius is 3 cm, then the