# 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 area as the sides. Find the dimensions that will minimize the cost. (Round your answers to three decimal places.)

1. 👍
2. 👎
3. 👁
1. what, no effort since you last [posted this problem? Most of it is just geometry.

If the radius is r and the cylinder length is h, then

4/3 π r^3 + πr^2 h = 1600
so
h = (4800 - 4πr^3)/(3πr^2)
= 1600/πr^2 - 4r/3

If the sides cost \$1/ft^2, then the ends cost \$2/ft^2, so the cost is

c = 2*4πr^2 + 2πrh
= 2*4πr^2 + 2πr(1600/πr^2 - 4r/3)
= 3200/r + 16π/3 r^2

so, to minimize cost, find where dc/dr = 0

dc/dr = -3200/r^2 + 32πr/3
= (32πr^3-9600)/3r^2
dc/dr=0 when
32πr^3 = 9600
r^3 = 300/π

r = 4.57
h = 18.28

1. 👍
2. 👎
2. total volume = cylinder + 2(hemispheres)
= πr^2 h + (4/3)π r^3
1600 = πr^2 h + (4/3)π r^3
4800 = 3πr^2 h + 4π r^3
h = (4800 - 4π r^3)/(3π r^2)

cost = 2(4π r^2) + 2π rh
= 8π r^2 + 2πr(4800 - 4π r^3)/(3π r^2)
= (16/3)π r^2 + 3200/r , not showing the simplification

d(cost)/dr = (32/3)πr - 3200/r^2
= 0 for a max/min of cost

32πr/3 = 3200/r^2
32π r^3 = 9600
r^3 = 9600/(32π) = 300/π
r = appr 4.57 ft
then
h = 18.283

state conclusion, but first check my arithmetic

1. 👍
2. 👎
3. *whew* what a relief that we agreed!!

1. 👍
2. 👎

## Similar Questions

1. ### physics

A string is wrapped around a uniform solid cylinder of radius r. The cylinder can rotate freely about its axis. The loose end of the string is attached to a block. The block and cylinder each have mass m. Note that the positive y

2. ### Physics

Four objects - a hoop, a solid cylinder, a solid sphere, and a thin, spherical shell - each has a mass of 4.59 kg and a radius of 0.252 m. (a) Find the moment of inertia for each object as it rotates about the axes shown in the

3. ### 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

4. ### algebera

1) Identify the solid form by the given net. :Triangular Prism :Triangular Pyramid*** :Cone :Triangle 2) Name the solid according to it's description: The figure has two bases that are parallel congruent circles. :Cylinder***

1. ### calculus

Find the maximum volume of right circular cylinder that can be inscribed in a cone of altitude 12 cm and base radius 4 cm, if the axes of the cylinder and con coincide.

2. ### Physics

A small metal cylinder rests on a circular turntable that is rotating at a constant speed. The small metal cylinder has a mass of 0.20kg, the coefficient of static friction between the cylinder and the turntable is 0.080, and the

3. ### Algebra

This diagram shows a solid metal cylinder. The cylinder hasbase radius 2x and height 9x. The cylinder is melted down and made into a sphere of radius r Find an expression for in terms of x

4. ### math

A container in the shape of a right circular cylinder with no top has surface area 3*pi (m2). What height h and base radius r will maximize the volume of the cylinder ?

1. ### Calculus

Find the moment of inertia of a right circular cylinder of radius of base R and height H, about the axis of the cylinder, if the density at the point P is proportional to the distance from P to the axis of the cylinder. Write down

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. ### maths

A rectangular sheet of paper 30 cm X 18cm can be transformed into a right circular cylinder in two ways - by rolling the paper along its length or by rolling it along its breadth . Find the ratio of the volume of two cylinders

4. ### Math

a 25.00 gram sample is placed in a graduated cylinder and then the cylinder is filled to the 50.0 mL mark with benzene. the mass of benzene and solid together is 58.80 gram. assuming that the solid is insoluble in benzene and that