You are given a 14 inch by 8 inch rectangular sheet of metal from which you are to construct a box. You are to cut a length, x, from each corner of the sheet of metal so that you can fold up the sides creating a box. Find the value of x that will yield the maximum volume of the box. Round your solution to the nearest hundredth.

x = 1.639 in

Step-by-step explanation:

With these instructions, the length of the box would be (14-2x), the width would be (8-2x), and the height would be x, so the equation for volume would be width*length*height, or

Now we solve for dV/dx because we want to find maximum volume

Now we let dV/dx equal zero to find critical values

,

Now we see where dV/dx changes signs:

dV/dx changes from positive to negative at

and from negative to positive at

so V(x) has a local maximum when x = in

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step-by-step explanation: