A rectangular box (base is not square) with an open top must have a length of 3ft, and a surface area of 16ft^2. Compute the dimensions of the box that will maximize its volume.

aanpalendmw 2022-07-18 Answered
Finding dimensions of a rectangular box to optimize volume
A rectangular box (base is not square) with an open top must have a length of 3ft, and a surface area of 16 f t 2 . Compute the dimensions of the box that will maximize its volume.
I am going wrong somewhere but I can't see where. I have set 2 ( 3 h + 3 w + h w ) = 16 and my optimization equation is v = h w l. I set w = 8 3 h 3 + h and plug into the optimization equation. Calculating v' gives me 9 h 2 84 h + 72 ( 3 + h ) 2 ,but setting that equal to zero does not give me the answer. I need for h, which is 1.
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Answers (2)

Abbigail Vaughn
Answered 2022-07-19 Author has 15 answers
Explanation:
The equation is going wrong you forgot to subtract the top so itll be 2 ( 3 b + b h + 3 h ) 3 b = 16 here b=breadth and h=height . Now you proceed by the way as you did and you will get an equation in h for volume diffrentiate it and put it to be 0. Thus we get eqn 48 h 18 h 2 3 + 2 h so on differentiating wrt h we get a quadratic as h 2 + 2 h 4 = 0 thus h = 1 + , 4 + 16 2 so we take positive sign for max area thus its 1 + 5 now you can get breadth and you are done with the sum.
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comAttitRize8
Answered 2022-07-20 Author has 2 answers
Step 1
Did you forget that the top of the box is open? The surface area is 2 ( 3 h + h w ) + 3 w = 16 , so that w = 16 6 h 3 + 2 h ..
Step 2
The volume of the box is V = 3 h w = 3 h 16 6 h 3 + 2 h .
If you differentiate this with respect to h, and set it equal to 0, you get the h for which the volume is maximized.
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