SOLUTION: A rectangular storage container with an open top is to have a volume of 12 cubic meters. The length of its base is twice the width. Material for the base costs 13 dollars per squar

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Question 1058007: A rectangular storage container with an open top is to have a volume of 12 cubic meters. The length of its base is twice the width. Material for the base costs 13 dollars per square meter. Material for the sides costs 7 dollars per square meter. Find the cost of materials for the cheapest such container.
Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
Let +h+ = the height of the box
Let +w+ = the width
+2w+ = the length
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The cost of the base is:
+2w%2Aw%2A13+
+26w%5E2+
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The area of the sides is:
+2%2Aw%2Ah+%2B+2%2A%28+2w+%29%2Ah+
+2w%2Ah+%2B+4w%2Ah+=+6w%2Ah+
The cost of the materials for sides is:
+6w%2Ah%2A7+=+42w%2Ah+
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Also given:
+w%2A2w%2Ah+=+12+
+2w%5E2h+=+12+
+h+=+12%2F%28+2w%5E2+%29+
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Let +C+ = cost of materials
The cost for materials is:
+26w%5E2+%2B+42w%2Ah+
+26w%5E2+%2B+42w%2A%28+12%2F%28+2w%5E2+%29+%29+
+C+=+26w%5E2+%2B+252%2Fw+
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I don't know if this is for a calculus course, but
that's the only way I can solve it.
Take the derivative and set = zero
+52w+-+252%2Fw%5E2+=+0+
+52w+=+252%2Fw%5E2+
+52w%5E3+=+252+
+w%5E3+=+4.8462+
+w+=+1.6923+
and
+C%5B+min+%5D+=+26w%5E2+%2B+252%2Fw+
+C%5B+min+%5D+=+26%2A1.6923%5E2+%2B+252%2F1.6923+
+C%5B+min+%5D+=+74.4573+%2B+148.9098+
+C%5Bmin%5D+=+223.367+
The minimum cost is $223.37
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Check:
Is the 2nd derivative positive? ( that makes it a min )
+52+-+252%2A%28-2%29%2Fw%5E3+
+52+%2B+504%2Fw%5E3+ yes, it's positive
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Here's the plot of +C%28w%29+:
+graph%28+400%2C+400%2C+-2%2C+10%2C+-400%2C+400%2C+26x%5E2+%2B+252%2Fx+%29+
I may have the right answer. Get another opinion on this