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Marina86 [1]
2 years ago
12

"a box is constructed out of two different types of metal. the metal for the top and bottom, which are both square, costs $4 per

square foot and the metal for the sides costs $7 per square foot. find the dimensions that minimize cost if the box has a volume of 30 cubic feet."
Mathematics
1 answer:
jarptica [38.1K]2 years ago
7 0

The base of the box is b ft by b ft and the height is h ft. 
<span>Since the volume is 40 ft³, </span>
<span>h = 40/b² </span>

<span>area of bottom = area of top = b² ft² </span>
<span>area of one side = bh </span>
<span>cost C = 2·4b² + 4·7bh </span>
<span>= 8b² + 28b(40/b²) </span>
<span>= 8b² + 1120/b </span>
<span>dC/db = 16b - 1120/b² </span>

<span>min cost when dC/db = 0 </span>
<span>16b = 1120/b² </span>
<span>b = ∛70 ≅ 4.12 ft </span>
<span>h = 40/b² ≅ 2.36 ft</span>
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