Exercise 28 Page 495

Let's take a look at the given drawing.

We are asked to determine the area of the yellow part of the tabletop. Looking at the drawing, we can see that the area of the yellow part is the area of the entire circle C minus the area of the red square S. Let's calculate the area of the entire circle first. Recall the formula for the area of a circle. A=π r^2Here, r is the radius of the circle. In our case r= 6x.

C=π r^2

Substitute

r= 6x

C=π ( 6x)^2

PowProdII

(a b)^m=a^m b^m

C=π* 6^2x^2

CalcPow

Calculate power

C=π* 36x^2

Multiply

Multiply

C=36π x^2

Next we will calculate the area of the red square. Recall the formula for the area of a square. A=s^2 Here, s is the side length of the square. In our case s= 3x.

S=s^2

Substitute

s= 3x

S=( 3x)^2

PowProdII

(a b)^m=a^m b^m

S=3^2x^2

CalcPow

Calculate power

S=9x^2

As we have mentioned before, the area of the yellow part A is equal to the area C minus the area S. A=C-S=36π x^2-9x^2 We have to rewrite our answer in factored form. Let's find the Greatest Common Factor (GCF) of the expression. To do it, we will factor each term in the expression. 36π x^2&=2* 2* 3* 3*π* x* x 9x^2&=3* 3* x* x The GCF is 3* 3* x* x, or 9x^2. Finally, we will factor out the GCF.

36π x^2-9x^2

▼

Factor out 9x^2

Rewrite

Rewrite 36π x^2 as 9x^2* 4π

9x^2* 4π-9x^2

Rewrite

Rewrite 9x^2 as 9x^2* 1

9x^2* 4π-9x^2* 1

FactorOut

Factor out 9x^2

9x^2(4π-1)

The factored form of the area of the yellow part of the tabletop is 9x^2(4π-1).