from a square piece of cardboard with length and width of x inches, a square of width x-1 inches is removed from the center. Write the area of the remaining piece of cardboard as a function of x
To find the area of the remaining piece of cardboard, we need to calculate the area of the whole square cardboard minus the area of the square that is removed from the center.
The area of the whole square cardboard is calculated by multiplying its length by its width, which in this case would be x inches. So, the area of the whole square cardboard is x * x = x^2 square inches.
The area of the square removed from the center can be calculated by multiplying its length by its width, which is (x-1) inches. So, the area of the square removed is (x-1) * (x-1) = (x-1)^2 square inches.
To find the area of the remaining piece of cardboard, we subtract the area of the square that is removed from the area of the whole square cardboard:
Area of the remaining piece = Area of the whole square cardboard - Area of the square removed
= x^2 - (x-1)^2 square inches
Expanding the square term (x-1)^2, we get:
Area of the remaining piece = x^2 - (x^2 - 2x + 1)
= x^2 - x^2 + 2x - 1
= 2x - 1 square inches
Therefore, the area of the remaining piece of cardboard is given by the function:
A(x) = 2x - 1, where x represents the width (and length) of the original square cardboard.