A rectangular storage container with an open top is to have a volume of 10 m3. The length of this base is twic?
A rectangular storage container with an open top is to have a volume of 10 m3. 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.
please tell me how to get to the answer
Since this is a rectangular container, we see that:
V = LWH.
Since L = 2W, we see that:
V = (2W)WH
==> V = 2W²H.
The area of the base is LW. So the cost for the production of the base is 10LW. The area of the sides are 2LW + 2WH; which makes a cost of 12LW + 12WH. So the total cost of the prism is:
C = 10LH + 12LW + 12WH
==> C = 10(2W)H + 12(2W)W + 12WH since L = 2W
==> C = 20WH + 24W² + 12WH
==> C = 24W² + 32WH.
Since V = 10, we see that:
10 = 2W²H
==> H = 5/W².
The cost in terms of width is:
C = 24W² + 32W(5/W²)
==> C = 24W² + 160/W.
Taking derivatives:
dC/dW = 48W – 320/W².
Setting dC/dW = 0:
48W – 320/W² = 0
==> 48W³ – 320 = 0
==> W ≈ 1.882 meters.
This implies that L = 2(1.882) = 3.764 meters and H = 5/(1.882)² = 1.412 meters.
Therefore, the minimum cost is:
C = 10LH + 12LW + 12WH = 10(3.764)(1.412) + 12(3.764)(1.882) + 12(1.882)(1.412) = $170.22.
I hope this helps!
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