An open box is to be made from a 10-ft by 14-ft rectangular piece of sheet metal by cutting out squares of equal size from the four corners and folding up the sides. what size squares should be cut to obtain a box with largest possible volume?
let each side of the cut-out-squares be x ft
so the box will be 10-2x y 14-2x by x
V = x(10-2x)(14-2x)
= x(140 -48x + 4x^2)
= 140x - 48x^2 + 4x^3
d(V)/dx = 140 - 96x + 12x^2 = 0 for a max of V
3x^2 - 24x + 35 = 0
x = (24 ± √156)/6
= appr. 1.92 or 6.08 , but obviously 10-2x > 0 ---> x > 5
cut out should be 1.92 ft by 1.92 ft
check my arithmetic
To find the size of the squares that should be cut to obtain a box with the largest possible volume, we can start by visualizing the situation.
First, let's draw a diagram to represent the rectangular piece of sheet metal:
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Next, we cut squares of equal size from each corner and fold up the sides to create an open box. The dimensions of the box will be the length of the squares cut from the corners, the width of the remaining piece of sheet metal, and the height of the folded-up sides.
Let's say we cut x feet from each corner.
After cutting the squares from the corners, the remaining piece of sheet metal will have dimensions of (10 - 2x) by (14 - 2x).
The height of the folded-up sides will be x.
So, the volume of the box can be calculated using the formula:
Volume = length * width * height
Volume = (10 - 2x)(14 - 2x)(x)
To find the size of squares that will result in the maximum volume, we need to maximize the volume function.
To do that, we can take the derivative of the volume function with respect to x and set it equal to zero to find the critical points.
Differentiating the volume function, we get:
dV/dx = -4x^3 + 48x^2 - 140x + 140
Setting dV/dx = 0, we can solve for x using various methods such as factoring, synthetic division or numerical methods like Newton's method.
Once we have the value(s) of x, we can substitute it back into the volume function to find the corresponding values of the volume.
Finally, we compare these volumes and choose the largest one.
Therefore, to find the size of squares that should be cut to obtain a box with the largest possible volume, we need to solve for x using the derivative of the volume function and find the value of x that maximizes the volume.