Question

Please answer both 13a) and 13b).

Answer 1

Answer:

Look below

Step-by-step 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:

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