Minimizing Surface Area in Tin Can Design: A Mathematical Analysis

[WiseGuy1995]

Homework Statement

Is minimiznig the area of tin used to make a can an important factor?
Suppose a manufacturer wishes to enclose a fixed volume,V, using a cylindrical can.
The height of the cylinder is denoted by h, and the radius of the cylinder can section by r.
i)Write a function for the surface area of the can.
ii)Determine what happens to the surface area as the radius increases.
iii)Determine what happens as the radius tends to zero.
iv)Find the values of r which minimizes the surface area.
v)Consider an alternative tin shape, with justifications, does your result support the argument that minimizing surface area is a key factor in the design of tin cans?

Homework Equations

Volume = πr^2h

The Attempt at a Solution

i)Surface Area = 2πr2 + 2πrh = 2πr(r+h)
ii)The surface area increases and the height decreases as the radius increases.

Confused about the rest.

 

[brocks]

The answer is 42.

 

[WiseGuy1995]

care to explain how you arrived at 42? thanks

 

[Mark44]

Homework Statement

Is minimiznig the area of tin used to make a can an important factor?
Suppose a manufacturer wishes to enclose a fixed volume,V, using a cylindrical can.
The height of the cylinder is denoted by h, and the radius of the cylinder can section by r.
i)Write a function for the surface area of the can.
ii)Determine what happens to the surface area as the radius increases.
iii)Determine what happens as the radius tends to zero.
iv)Find the values of r which minimizes the surface area.
v)Consider an alternative tin shape, with justifications, does your result support the argument that minimizing surface area is a key factor in the design of tin cans?

Homework Equations

Volume = πr^2h

The Attempt at a Solution

i)Surface Area = 2πr2 + 2πrh = 2πr(r+h)
ii)The surface area increases and the height decreases as the radius increases.

Confused about the rest.

You also know that the volume V is fixed, so you can solve for r as a function of h and the fixed V.

 

[WiseGuy1995]

do you mean like 2πr^2 + 2πr(V ÷ πr^2) and differentiate?

 

[Dick]

do you mean like 2πr^2 + 2πr(V ÷ πr^2) and differentiate?

Yes, now you have the surface area in terms of the single variable r. Now you can minimize it using derivatives.

 

[WiseGuy1995]

when i did the differentiation i got r^3 = 2V/4π

Is this correct?

 

[Dick]

when i did the differentiation i got r^3 = 2V/4π

Is this correct?

That agrees with what I got as the answer to iv). Though you haven't actually solved for r yet.

 

[WiseGuy1995]

That agrees with what I got as the answer to iv).

iv) i thought i was answering iii)
so how do i get the answer for iii) ?

 

[Dick]

iv) i thought i was answering iii)
so how do i get the answer for iii) ?

You gave an answer to a question. If the question is "iii)Determine what happens as the radius tends to zero." I don't think "r^3 = 2V/4π" makes much sense as an answer. If the question is "iv)Find the values of r which minimizes the surface area." Then I think it does. Are you trying to confuse me?

 

[WiseGuy1995]

yea sorry about that i now understood that i actually answered iv)
not trying to confuse you , just a lapse in concentration on my part.
thanks for your help btw

 

[brocks]

care to explain how you arrived at 42? thanks

When I responded to the OP, it was just the blank homework form. You must have come back and filled it in later.