Exercise 27 Page 495

Let's draw the mirror described in the exercise.

We are asked to calculate the area of the metal frame. We can see that the area of the frame is the area of the entire square, F, minus the area of the circular mirror, M. Let's calculate the area of the entire square first. Recall the formula for the area of a square. A=s^2Here, s is the side length of the square. In our case s= 15x.

F=s^2

Substitute

s= 15x

F=( 15x)^2

PowProdII

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

F=15^2x^2

CalcPow

Calculate power

F=225x^2

Next we will calculate the area of the circular mirror. Recall the formula for the area of a circle. A=π r^2 Here, r is the radius of the circle. In our case r= 5x.

M=π r^2

Substitute

r= 5x

M=π ( 5x)^2

▼

Simplify

PowProdII

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

M=π* 5^2x^2

CalcPow

Calculate power

M=π* 25x^2

Multiply

Multiply

M=25π x^2

As we have mentioned before, the area of the frame A is equal to the area F minus the area M. A=F-M=225x^2-25π x^2 We have to rewrite our answer in factored form. Let's find the Greatest Common Factor (GCF) of the expression. To do so, we will factor each term in the expression. 225x^2=&3* 3* 5* 5* x* x 25π x^2=& 5* 5*π* x* x The GCF is 5* 5* x* x, or 25x^2. Finally, we will factor out the GCF.

225x^2-25π x^2

▼

Factor out 25x^2

Rewrite

Rewrite 225x^2 as 25x^2* 9

25x^2* 9-25π x^2

Rewrite

Rewrite 25π x^2 as 25x^2*π

25x^2* 9-25x^2*π

FactorOut

Factor out 25x^2

25x^2(9-π)

The factored form of the area of the metal frame is 25x^2(9-π).