Finding Cost of Cylinder with Constant Radius V

[robrob]

Homework Statement

How do I show that when I have C = 4√(3)r^2 + 2π(r)h + k(4π(r) + h), the cost C to make a cylinder of constant radius V gives the following defining equation: (∛(V))/k = (∛(π(h)/r)) x (2π - h/r)/π(h/r) - 4√(3)

k is the reciprocal of the length that can be joined for the cost of one unit area of metal.

Homework Equations

((h/r) = 8/π ≈ 2.55 is the minimized amount of metal used

The Attempt at a Solution

I've tried substituting h = 1000/(πr^2) and then found the derivative with respect to r but it doesn't prove the equation above.
We haven't learn partial derivatives yet, so is there any other way to solve this?

 

[mighty2000]

Thank you for your question. I would like to provide you with some guidance on how to approach this problem.

Firstly, it is important to understand the given equations and variables. The equation C = 4√(3)r^2 + 2π(r)h + k(4π(r) + h) represents the total cost (C) to make a cylinder of constant radius (r) with a given height (h) and a constant cost factor (k). The variable V represents the volume of the cylinder, and k is the reciprocal of the length that can be joined for the cost of one unit area of metal.

To prove the given defining equation, we need to show that when we substitute the values of h and r from the homework equations ((h/r) = 8/π and (∛(V))/k = (∛(π(h)/r)) x (2π - h/r)/π(h/r) - 4√(3)), we get the same result on both sides of the equation.

One way to approach this problem is by using the given equations to express h in terms of r and V. Then, substitute this value of h in the defining equation and simplify it to get the same result on both sides. I suggest you try this method and see if it helps you prove the equation.

If you are still unable to solve the problem, you can try reaching out to your instructor or classmates for help. Additionally, you can also consult some online resources or textbooks for guidance on solving similar problems.

I hope this helps. Good luck with your assignment!