Question

100 Points! A box without a top is to be made to hold copious bags of coffee. It is made from a rectangular piece of cardboard, with dimensions 4.5 ft by 3 ft, by cutting out square corners with side length x and folding up the sides.a) Write an equation for the volume V of the box in terms of x.

(b) Use Desmos to graph and estimate the value of x, to the nearest tenth, that gives the greatest volume. Explain your process. Post a snapshot of your graph with your work.
(c) Explain what the x- and y-coordinates represent in your equation.

Answer 1

Let the length of the side of the square being cut out equal x

The length of the box would be 4.5 - 2x

The width of the box would be 3 -2x

Volume = (4.5 -2x) * (3-2x) * x

Simplify to:

V = 4x^3 - 15x^2 + 13.5x

B See picture: The greatest volume would be the point of the highest curve.

x = 0.589 y = 3.565, Rounded to the nearest tenth x = 0.6

Process: entered the equation from A into Desmos. The Y value would be the volume, so found where the volume was the highest and then found the related x value.

C) X is the side length of the corner squares being cut out, which would also be the height of the box. The Y value is the volume of the box.

Answer 2

Let the length of the side of the square being cut out equal x

The length of the box would be 4.5 - 2x

The width of the box would be 3 -2x

Volume = (4.5 -2x) * (3-2x) * x

Simplify to:

V = 4x^3 - 15x^2 + 13.5x

B See picture: The greatest volume would be the point of the highest curve.

x = 0.589 y = 3.565, Rounded to the nearest tenth x = 0.6

Process: entered the equation from A into Desmos. The Y value would be the volume, so found where the volume was the highest and then found the related x value.

C) X is the side length of the corner squares being cut out, which would also be the height of the box. The Y value is the volume of the box.