Emerson is making a box without a top from a rectangular piece of cardboard, with dimensions 12in by 16in, by cutting out square corners with side length x in. A) Write an equation for the volume V of the box in terms of x. B) Use technology to estimate the value of x that gives the greatest volume. Round the value to the nearest tenth. C) Assume Emerson used the value of d you found in Part (b) to make his box. What were the dimensions of Emerson’s box?

1 answer
Question:

Emerson is making a box without a top from a rectangular piece of cardboard, with dimensions 12in by 16in, by cutting out square corners with side length x in.
A) Write an equation for the volume V of the box in terms of x.
B) Use technology to estimate the value of x that gives the greatest volume. Round the value to the nearest tenth.
C) Assume Emerson used the value of d you found in Part (b) to make his box. What were the dimensions of Emerson’s box?

Answers

Answer:Step-by-step explanation:The volume of a rectangular box is width times length times height:V = wlhAfter the cardboard is folded, the width is 12 - 2x, the length is 16 - 2x, and the height is x.So the volume is:V = (12 - 2x) (16 - 2x) xIf we graph this, we get a wave: desmos.com/calculator/rsjosgzuxzThe wave is the highest at around x = 2.3 in.If we set x = 2.3:w = 12 - 2x = 7.4l = 16 - 2x = 11.4h = x = 2.3

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