## An open box is to be made from a six inch by six inch piece of material by cutting equal squares from the corners and turning up the sides.

Question

An open box is to be made from a six inch by six inch piece of material by cutting equal squares from the corners and turning up the sides. Find the volume of the largest box that can be made.

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1 month 2021-10-15T01:55:59+00:00 2 Answers 0 views 0

8 inch³

Step-by-step explanation:

Side should be 2 inch

Volume = side³ = 2³ = 8 inch³

Step-by-step explanation: So this is a fun little calculus optimization problem. First, we must make an equation describing the volume of the box. Each side is 6 inches starting and then we remove “x” from each side twice. Both sides can then be described as 6 – 2x. Then the height of the box will be “x” inches. Our total equation is then V = (6-2x)^2 • x. Lets FOIL this equation out.

(6-2x)(6-2x)•x = (4x^2 – 24x + 36) • x

Then multiply the x

V = 4x^3 – 24x^2 + 36x

So this equation describes the volume. We want to find the maximum volume possible. This will be at the highest point of the graph where the slope = 0. So we take the derivative.

dy/dx 4x^3 – 24x^2 + 36x = 12x^2 – 48x + 36.

Now we plug this into the quadratic formula to find the points where the slope (x) equals zero.

x = (48 +- )/24

sqrt(2304 – 1728) = sqrt(576) = 24

So x = (48 +- 24)/24

Our two answers are (48-24)/24 = 1 = x

and (48+24)/24 = 3 = x

These two answers will be our minimum and maximum values for volume.

Now we can think, if we take 3 inches from the sides of the box and we have two sides, there will be no volume, so that can’t be correct. And we can test this by plugging 3 into the equation. 4(3)^3 – 24(3)^2 + 36(3) = 0.

So 1 must be the maximum. When we plug this into the equation we get

4(1)^3 – 24(1)^2 + 36(1) = 16.

So the answer is 16in^3 by taking 1 inch squares out. This makes each side 4 inches and the height 1 giving us the volume equation. 4in•4in•1in = 16 in^3.

Please comment if anything needs clarified.