Calc help? ?

Question

I'm in project based calc II and I'm been working on this problem, but can't seem to get it right.

This is the problem:

A rectangular storage container with an open top is to have a volume of 10 m^3. The length of this base is twice the width. Material for the base costs $10 per square meter. Material for the sides costs $6 per square meter. Find the cost of materials for the cheapest such container.

I keep getting $128.66 for my answer even when I did it two separate ways. Can anyone help me. Please show work, so that I can learn how to do it myself, thanks!

Answer 1

Let x = width

=> 2x = length

Let h = height

=> 2x² h= 10

=> x² h= 5

Cost,

C = 10*2x² + 6*(6xh)

=> C = 20x² + 36x * (5/x²) = 20x² + 180/x

For C to be minimum, dC/dx = 0 and d²C/dx² > 0

dC/dx = 0

=> 40x - 180/x² = 0

=> x^3 = 4.5

=> x = 1.65 m width and 3.3 m length and 1.84 m height

Volume = (1.65)(3.3)(1.84) = 10.0 m³

d²C/dx² = 40 + 360/x³ > 0 => C is minimum.

Minimum cost of materials

= 20x² + 180/x

= 54.45 + 109.10

= $ 163.55.

Answer 2

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