Mrs. Lindsay doesn't have a cat crate and needs to take her cats to the vet. She's going to make a box out of a 8 foot by 8 foot piece of cardboard. How many feet
should Mrs. Lindsay cut from the corners to have the maximum amount of space
inside the box?

The amount to remove from each corner is a square, 4/3 ft by 4/3 ft.

Step-by-step explanation:

The piece of cardboard measures 8 ft by 8 ft.

Each corner will be cut to create the sides of the box.

Let the square cut from each corner measure x by x ft.

The base of the box will measure 8 - 2x by 8 - 2x, and the height of the box will be x.

V = (8 - 2x)^2 * x

V = 64x - 32x^2 + 4x^3

Now we differentiate and set the derivative equal to zero to find a maximum value.

64 - 64x + 12x^2 = 0

3x^2 - 16x + 16 = 0

(3x - 4)(x - 4) = 0

3x - 4 = 0  or  x - 4 = 0

3x = 4   or   x = 4

x = 4/3  or  x = 4

The solution x = 4 must be discarded since 8 - 2x = 8 - 8 = 0, and the sides would have 0 length.

The amount to remove from each corner is a square, 4/3 ft by 4/3 ft.

when rotating a figure 180 degrees, imagine that you are able to take the figure and turn it clockwise or counterclockwise around a center point. to rotate a figure 180 degrees, you will need to apply the rule (x, y) → (-x, -y). start by using a coordinate grid with coordinates for each vertex of the figure. the center point of the coordinate grid is located at (0, 0), which is what you will rotate the figure around. write down the original coordinates of the shape you are going to rotate. then, apply the rule. for example, coordinate (1, 2) becomes (-1, -2). another example would be, coordinate (-4, -2) becoming (4, 2). after you change each original coordinate to the rotated coordinate, you will draw your new figure.

step-by-step explanation: