A rectangular storage container with an open top is to have a volume of 16 cubic meters. The length of its base is twice the width. Material for the base costs 11 dollars per square meter. Material for the sides costs 5 dollars per square meter. Find the cost of materials for the cheapest such container. Round to the nearest penny.

Question
Asked Nov 22, 2019
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A rectangular storage container with an open top is to have a volume of 16 cubic meters. The length of its base is twice the width. Material for the base costs 11 dollars per square meter. Material for the sides costs 5 dollars per square meter. Find the cost of materials for the cheapest such container. Round to the nearest penny.

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Expert Answer

Step 1
The volume of the box is 16 cubic meters.
Let w be the width of the container, the length / is 2w
Note that, the volume of the rectangular container is V = 1x wxh
(1=2w)
V = 2w xw x h
(V16)
16 2w2h
16
h
2w2
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The volume of the box is 16 cubic meters. Let w be the width of the container, the length / is 2w Note that, the volume of the rectangular container is V = 1x wxh (1=2w) V = 2w xw x h (V16) 16 2w2h 16 h 2w2

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Step 2
Obtain the cost function of the material
Cost(C(w)) 11x area of the base + 5x area of the sides
=11x(1x w)5(2x 1xh+ 2x wx h)
16
2x w x
2w2
16
11(2w252x2wx-
2w2
240
C(w) 22w2
240
with respect to w
Differentiate C(w) =22w2 +
W
240
22w2
C'(w)
dw
240
=44w
44w3-240
help_outline

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Obtain the cost function of the material Cost(C(w)) 11x area of the base + 5x area of the sides =11x(1x w)5(2x 1xh+ 2x wx h) 16 2x w x 2w2 16 11(2w252x2wx- 2w2 240 C(w) 22w2 240 with respect to w Differentiate C(w) =22w2 + W 240 22w2 C'(w) dw 240 =44w 44w3-240

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Step 3
Equate the derivative to zero to find the critical point.
44w3240
= (0
44w3240 0
44w3 240
240
44
w = 1.76
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Equate the derivative to zero to find the critical point. 44w3240 = (0 44w3240 0 44w3 240 240 44 w = 1.76

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