# calculus

An open box of maximum volume is to be made from a square piece of cardboard, 24 inches on each side, by cutting equal squares from the corners and turning up the sides to make the box.
(a) Express the volume V of the box as a function of x, where x is edge length of the square cut-outs.
(b) What are the dimensions of the box that enclose the largest possible volume? State your answer in the form length by width by height.
(c) What is the maximum volume?

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1. let each side of the equal squares be x inches
length of box = 24-2x
width of box = 24-2x
height of box = x

a) Volume = x(24-2x)(24-2x)

b) expand the volume equation, then take the first derivative.
Set that derivative equal to zero. You will have a quadratic equation. Take the positive answer which lies between 0 and 12

c) put the answer from b) into the volume equation and evaluate.

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2. 20>o

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3. An open box is to be made from a 24 cm square cardboard by cutting equal squares out of the corners and turning up the sides. What is its optimization equation to find the height of the box that will give a maximum volume?

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