1. You are cutting colored cardboard rectangles for school art show. Each art piece will be placed on one of the rectangles so the outside of the rectangle forms a border or mat for the art piece. You are to cut the cardboard to fit artwork that is 3 times longer than it is wide. The mat should be 2 inches wide on each side. Write a polynomial expression for the area of cardboard rectangle. Simplify the expression.
Thanks!
I assume you want the area of the exposed part of the cardboard rectangle
it would be
3x^2 - (x-4)(3x-4)
= 3x^2 - (3x^2 - 16x - 16)
= 16x + 16
To find the area of the cardboard rectangle, we need to determine the dimensions of the rectangle first.
Let's assume the width of the artwork is "x" inches. Since the artwork is 3 times longer than it is wide, the length of the artwork would be 3x inches.
Now, let's calculate the dimensions of the cardboard rectangle. Since we want to create a 2-inch wide border/mat on each side of the artwork, the width of the cardboard rectangle would be (x + 2 + 2) inches, which simplifies to (x + 4) inches. Similarly, the length of the cardboard rectangle would be (3x + 2 + 2) inches, which simplifies to (3x + 4) inches.
To calculate the area of a rectangle, we multiply the length by the width. Therefore, the polynomial expression for the area of the cardboard rectangle would be:
A = (x + 4)(3x + 4)
To simplify this expression, we can use the distributive property:
A = 3x(x) + 4(x) + 4(3x) + 4(4)
A = 3x^2 + 4x + 12x + 16
A = 3x^2 + 16x + 16
So, the simplified polynomial expression for the area of the cardboard rectangle is 3x^2 + 16x + 16.