Question

Please answer both 13a) and 13b).

Answer 1

Look below

Explanation:

(a) The area of the shaded region is the area of the larger rectangle minus the area of the smaller rectangle.

Start by finding the area of the larger rectangle:

`A=lw\\A=4x*8x\\A=32x^2`

Then, find the area of the smaller rectangle:

`A=lw\\A=4y*2y\\A=8y^2`

Therefore the area of the shaded region is:

`32x^2-8y^2`

Note that we still need to factor it...

`8(4x^2-y^2)`

`4x^2-y^2\ \text{can be written as a difference of squares}`

`a^2-b^2=(a-b)(a+b)`

`4x^2-y^2=(2x)^2-(y)^2=(2x-y)(2x+y)`

Therefore the fully factored form is:

`8(2x-y)(2x+y)`

(b) The area of the shaded region is the area of the larger circle minus the area of the smaller circle.

Start by finding the area of the larger circle:

`A=\pi r^2\\A=\pi R^2`

The area of the smaller circle is:

`A=\pi r^2\\`

Therefore the difference is:

`\pi R^2-\pi r^2`

We can factor out
`\pi`

`\pi(R^2-r^2)`

Note this can again be written as a difference of squares:

`(R)^2-(r)^2=(R-r)(R+r)`

Therefore the fully factored form is:

`\pi(R-r)(R+r)`