Posted by **Martina** on Thursday, March 29, 2007 at 11:59pm.

A rectangular storage container with an open top is to have a volume of 15m^3. The length of the base is twice the width of the base. Material for the base costs $6 per square meter. Material for the sides costs $7 per square meter. Find the cost of the materials for the cheapest such container. Round answer to nearest hundredth.

I know that you need to find an equation, get the derivative, find the critical numbers, and then find the minimum. But, I have no clue on how to find the equation! If someone could please help me!

You will need a diagram.

Visualize the box "flattened", you have a rectangular base, with rectangles on each of its sides.

These four rectangles must have the same width, namely the height of the box.

Let that height be h m.

Let the width of the base be x m, then its length is 2x m.

So the volume is x(2x)h or 2hx^2 m^3

but we know this is 15.

So one of the equations is 2hx^2 = 15 or

h = 7.5/x^2

Since you want to minimize the Cost, you now need an equation for Cost

Cost of base = 6(x)(2x)=12x^2

Cost of sides = 2(7)(x)h) + 2(7)(2x)(h)

=42xh

Cost = 12x^2 + 42xh

=12x^2 + 42x(7.5/x^2)

=12x^2 + 315/x

Cost' = 24x - 315/x^2 = 0 for a min of Cost

Solve this....i got x=2.3588....

You actually have to plug it into the original cost equation and I get $200.31 and it's the right answer! Thank you so very much for your help!!!!

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