Answer :
As per area of rectangle, the dimensions of the box that yield the lowest cost is (1,1,6)
Area of rectangle:
The standard formula for area of rectangle is calculated as,
A = length x breadth
Given,
A closed rectangular box with volume 6 cubic feet is made from two different types of cardboard. the top and bottom are made with a heavy-duty cardboard that costs 30 cents per square foot. The sides are made from a light-weight cardboard that costs 5 cents per square foot.
Here find the dimensions of the box that yield the lowest cost.
Here let's consider the dimensions of the rectangular box be length l, breadth is b, and height is h.
Now, from the given information, the total cost can be derived as,
=> C=2lb×48+(2bh+2lh)×8
=> C=96lb+16bh+16lh
=> C=16(6lb+bh+lh) -----------------→(1)
Then the given volume is, V = lbh = 6 ---------------→(2) cubic feet.
Then the equation 2,
=> lbh = 6h = 6/lb
And then we have to substitute the value of h in equation 1, we get
=> C(l,b,h)=16(6lb+6/l+6/b)
Now we need to minimize the resulting function of two variables.
Therefore, we need to partially derivative the above function with respect to l,b. And set these equal to zero.
And the for l=0,b is undefined. So we can eliminate l=0.
Here we have to substitute l=1,b=112⇒1 in equation 2, we get
=> lbh=6(1)(1)h=6h=6h=6
Therefore, the critical point is, (1,1,6)
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