SOLUTION: An engineer needs a metal box to shield sensitive electronic devices from external electric fields. One side of the box should be open so that it can be placed over the components.
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Question 1071558: An engineer needs a metal box to shield sensitive electronic devices from external electric fields. One side of the box should be open so that it can be placed over the components. The box can be made from a 3 m x 4 m sheet of metal by cutting squares from the corners and folding up the sides.
1. What is the maximum volume of the box?
2. What are the box's dimensions?
I already tried to solve for it using the equation (4-2x)(3-2x)(x) expanding it, trying to solve for zeros but nothing is working. Please help. Found 2 solutions by ankor@dixie-net.com, ikleyn:Answer by ankor@dixie-net.com(22740) (Show Source):
You can put this solution on YOUR website! The box can be made from a 3 m x 4 m sheet of metal by cutting squares from the corners and folding up the sides.
:
One way is t0 plot the equation y = x(4-2x)(3-2x), y being the volume
1. What is the maximum volume of the box?
Max occurs when x=.566 inches, then y = 3.032 cu/m
:
2. What are the box's dimensions?
Height = .566 m
4 - 2(.566) = 2.87 m is the length
3 - 2(.566) = 1.87 m is the width
The volume is V(x) = x*(4-2x)*(3-2x) = x*(4x^2 -2x*(4+3) + 12)) = x*(4x^2 - 14x + 12) = 4x^3 - 14x^2 + 12x.
Take the derivative:
V'(x) = 12x^2 - 28x + 12
Make it equal to zero:
12x^2 - 28x + 12 = 0,
3x^2 - 7x + 3 = 0,
= = .
= 1.768 (approximately), and it is clear that this root doesn't work.
= 0.566 (approximately.
Answer. The volume is maximal at x = 0.566 m.
This solution is pretty straightforward and should not be difficult for you.
