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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