A cylindrical can without a top must be constructed to contain V=2100cm3 of liquid. Find the base radius r and height h that will minimize the amount of material needed to construct the can.

Question

I am just completely lost on this question. I am a first year student trying to take calc one online and I am completely struggling.

Answer 1

Quontity of material represent here surface area S = 2πrh + πr².

We have to find minimum of this function under condition that volume V = πr²h = 2100. From here h = 2100/(πr²). Then S = 2πr·2100/(πr²) + πr² = 4200/r + πr². We get S as function of one variable.

Derivative S' = - π4200/r² + 2πr = 0, 2πr.³ = 4200, r³ = 2100/π.From here critical number is r = (2100/π)^1/3 ≈ 8.7436 cm. This is will be minimum because by second derivative test S''(r) = 8400/r³ + 2π > 0 at critical number r = (2100/π)^1/3. Let found coreesponding h = 2100/(π(2100/π)^2/3 =(2100/π)^1/3≈ 8.7436 cm

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Answer 2

Step 1: Determine what they are asking

They want the material to be minimized. That is equivalent to asking for the surface area to be minimized. So the basic function we need to use is a surface area function (of a cylinder).

Surface Area (SA) = 2πrh + πr² (normally it would be 2πr² but they say it doesn't have a top)

In Calculus, the way we minimize a function is to take the derivative of it and set it equal to zero. But, we have 2 variables in this function ("r" and "h" so we need to figure out how to eliminate one of these.

Step 2: Find a relationship between the variables so you can write one in terms of the other to eliminate it in the function.

Th volume of a cylinder (V) = πr²h. In this case V is 2100 so we can write 2100 = πr²h which we can now solve for h to get: h = 2100/(πr²).

Step 3: Write the Basic equation in one variable:

SA(r) = 2πr[2100/(πr²)] + πr²

SA(r) = 4200/r + πr²

Step 4: Take the derivative, set it equal to zero, and solve

SA'(r) = -4200/r² + 2πr

0 = -4200/r² + 2πr

4200/r² = 2πr

r³ = 4200/(2π)

r³ = 668.45 cm

r = 8.74 cm

Using the relationship we had between h and r,

h = 2100/(πr²)

h = 2100/(π8.74²)

h = 8.74 cm