Gloria would like to construct a box with volume of exactly 55ft3 using only metal and wood. The metal costs $7/ft2 and the wood costs $5/ft2. If the wood is to go on the sides, the metal is to go on the top and bottom, and if the length of the base is to be 3 times the width of the base, find the dimensions of the box that will minimize the cost of construction. Round your answer to the nearest two decimal places.

The dimensions of the box are:

W=2.97 ft

L = 8.91ft

H=2.08ft

Step-by-step explanation:

Volume of the box= 55ft³

LWH=55

Length of the base, L = 3W

3W*WH=55

3W²H=55

Metal costs =$7/ft²

Wood costs =$5/ft².

Area of the Sides= 2(LH+WH)

=

Cost of the sides therefore is:

Area of the top and bottom=2LW=2*3W*W=6W²

Cost=7*6W²=42W²

Total Cost, C(W) of the box:

To find the minimum cost, we set the derivative of C(W) to be equal to zero.

84W³=2200

W³=26.19

W=2.97 ft

L = 3W=3*2.97=8.91ft

The dimensions of the box are:

W=2.97 ft

L = 8.91ft

H=2.08ft

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step-by-step explanation:

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