A rectangular storage container with an open top is to have a volume of 10 cubic

Question

A rectangular storage container with an open top is to have a volume of 10 cubic meters. The length of its base is twice the width. Material for the base costs 10 dollars per square meter. Material for the sides costs 4 dollars per square meter. Find the cost of materials for the cheapest such container Total cost = (Round to the nearest penny and include monetary units. For example, if your answer is 1.095, enter $1.10 including the dollar sign and second decimal place.) A fence is to be built to enclose a rectangular area of 230 square feet. The fence along three sides is to be made of material that costs 3 dollars per foot, and the material for the fourth side costs 13 dollars per foot. Find the length L and width W (with W leq L) of the enclosure that is most economical to construct. L = W =

Explanation / Answer

1.

Picture a rectangle with area of 230 and sides of L and W

so L x W = 230
and L = 230 / W

For the cost analysis

L = $3/ft and W = $3/ft and $13/ft

so 2 * L(ft) * $3/ft + 1 * W(ft) * $3/ft + 1*W(ft) * $13/ft = cost of the fence

substitute L = 230 / W
2 * (230 / W) * 3 + 3 * W + 13 * W = cost

To find the lost cost take the derivative of the left hand side and set it equal to zero and solve
-1380 / W^2 + 16 = 0
W = 9.29 ft

Solve for L
L = 230 / W = 230 / 9.29
L = 24.77 ft

Now substitute the values for W and L into the initial cost equation
Cost = 2 * L * 3 + 3 * W + 13 * W
Cost = 2 * 24.77 * 3 + 3 * 9.29 + 14 * 9.29
Cost = 148.6 + 157.92
Cost = $306.5

2.
Since this is a rectangular container, we see that:
V = LWH.
Since L = 2W, we see that:

V = (2W)WH
==> V = 2W

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