Shaded area question

Question

1. Write an expression for the area of the shaded outer region in factored form. (4 marks)

Please answer in full detail

Inner region variables:2y length. y height

Outer shaded variables: X, X, X, X

Bothe the outer and inner area are rectangles. The outer area is larger then the inner area

Answer 1

Could also break the shaded portion into 4 pieces whose areas are:

x(2y + 2x) top piece

xy + xy the two pieces on either side of the inside rectangle

x(2y + 2x) bottom piece (same as top piece).

Add them: 2x(2y + 2x) + 2xy = 4xy + 4x² + 2xy = 4x² + 6xy. Factored: 2x(2x + 3y)

Answer 2

If you assume that the x dimension is the length from the inside rectangle to the outer, then your length of the outer will be 2y+2x and the length of the inner will be y+2x. The area of the outer then becomes A_outer=(2y+2x)(y+2x)=2y²+4xy+2xy+4x²=2y²+6xy+4x².

The area of the shaded region is A_outer-A_inner. The inner area was calculated to be 2y²

A_Shaded=2y²+6xy+4x²-2y²=6xy+4x²

After factoring a 2x out of the expression, the final answer will be A_Shaded=2x(3y+2x)