,,, A rectangular box with a volume of 64 ft3 is to be constructed with a square
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,,,
A rectangular box with a volume of 64 ft3 is to be constructed with a square base and top. The cost per square foot for the bottom is 15 , for the top is 10 , and for the sides is 2.5 . What dimensions will minimize the cost? What are the dimensions of the box? The length of one side of the base is The height of the box is (Round to one decimal place as needed.) A mailing service places a limit of 36 in. on the combined length and girth of (distance around) a package to be sent parcel post. What dimensions of a rectangular box with square cross-section will contain the largest volume that can be mailed? (Hint: There are two different girths.) The dimensions are x = and y =Explanation / Answer
1
Volume of box = 64 = x^2*y
y = 64/x^2.....(1)
Total cost = 4*(xy)*10 + (x*x)*15 + (x*x)*10 = 10xy + 25x^2...........(2)
Using 1 and 2
Total cost = 640/x + 25x^2
f(x) = 25x^2 + 640/x
Now minimising f(x) we get x = 2.34
Hieght of box = 11.69
2
x + y = 36
y = 36-x
Volume = x^2*y = x^2*(36-x) = 36x^2 -x^3
f(x) = -x^3 +36x^2
Now maximise f(x) and we get
x = 24 and y =12
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