Elementary Analysis
posted by JEZRELL .
A company wants to manufacture cylindrical aluminum cans with a value of 1000cm^3 (1 Liter) What radius and height of the can be to minimize the amount of aluminum used? Please help! This is very hard. Thanks.

You know that
pi r^2 h = 1000, so
h = 1000/(pi r^2)
the surface area is
a = 2pi r^2 + 2pi r h
= 2pi r^2 + 2pi r * 1000/(pi r^2)
= 2pi r^2 + 2000/r
da/dr = 4pi r  2000/r^2
= (4pi r^3  2000)/r^2
Since r > 0, da/dr=0 when
4pi r^3 = 2000
r^3 = 2000/4pi = 500/pi
r = 5 ∛(4/pi) ~= 5.42
go to wolframalpha.com and enter
plot y = 2pi r^2 + 2000/r where 3<r<8
to see where it is minimum
Respond to this Question
Similar Questions

math
An aluminum cylindrical juice can has a height of 4 in. and a radius of 6 in. How many square inches of aluminum are needed to make the can? 
math
you are designing a 1000cm^3 right circular cylindrical can whose manufacture will take waste into account. There is no waste in cutting the aluminum for the side, but the top and bottom of radius r will be cut from squares that measure … 
Math
An aluminum cylindrical juice can has a height of 4 in. and a radius of 6 in. How many square inches of aluminum are needed to make the can? 
Chemistry
Aluminum is more reactive than iron, yet it is used today for a variety of applications in which iron would corrode (cans, rain gutters,etc). The reason for the corrosion durability of aluminum is 1. all aluminum products are treated … 
chemistry
an aluminum cylinder is 10.0 cm in length and has a radius of .25 cm. if the mass of a single atom of aluminum is 4.48X10^23 calculate the number of aluminum atoms present in the cylinder. the density of aluminum is 2.70 g/cm^3 Please … 
problem solving
a pound of aluminum may be used to make 29 aluminum cans. there are 240 pounds of aluminum in a 3,000 pound car. how many aluminum cans could be made from the aluminum in a,3,000 pound car? 
Calculus1
You are designing a 1000 cubic cm closed cylindrical can with radius r and height h. Assume that there is no waste in cutting the aluminum for the side of the can, but that the circular top and bottom are cut from square sheets of … 
calculus
a. A closed cylindrical can is to hold 1000cm^3 of liquid. How should we choose the height and radius of this can to minimize the amount of material need to manufacture the can? 
calculus
A closed cylindrical can is to hold 2 litter (200 cmÂ³) of liquid. How should we choose the height and radius to minimize the amount of material needed to manufacture the can? 
Algebra
Your class project involves recycling aluminum cans. After x weeks, your class has (13x + 50) aluminum cans. The class goal is to collect (80x + 120) aluminum cans. How many more aluminum can does your class need to collect?