Exercise 21 Page 88

Consider that three square mirrors are used for a light reflection experiment. Since all mirrors are square, we know that Mirror A is similar to Mirror B and Mirror B is similar to Mirror C. This means that Mirror A is also similar to Mirror C. IfA ~ B and B ~ C, then A ~ C We want to find the area of Mirror A. To do so, remember that when two figures are similar, the value of the ratio of their areas is equal to the square of the value of the ratio of their corresponding sides lengths. In our case, the ratio of the corresponding sides for Mirrors A and B is equal to 5:6. A_A/A_B=(5/6)^2 This ratio shows the relationship between the areas of Mirror A and Mirror B. Even though we do not know the area for Mirror B yet, let's go ahead and solve this equation for the area of Mirror A, A_A.

A_A/A_B=(5/6)^2

PowQuot

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

A_A/A_B=5^2/6^2

CalcPow

Calculate power

A_A/A_B=25/36

MultEqn

LHS * A_B=RHS* A_B

A_A/A_B* A_B=25/36* A_B

FracMultDenomToNumber

a/A_B* A_B = a

A_A = 25/36* A_B

As we can see, the area of Mirror A depends on the area of Mirror B. Let's find the area of Mirror B!

Area of Mirror B

We know that the ratio of the area of Mirror B to the area of Mirror C is 16:25. We can use the formula for the ratio between areas again to help us find the area of Mirror B. A_B/A_C=16/25 Let's solve this equation for A_B. Notice that since we do not yet know the area of Mirror C, we will end up with another expression.

A_B/A_C=16/25

MultEqn

LHS * A_C=RHS* A_C

A_B/A_C* A_C=16/25* A_C

FracMultDenomToNumber

a/A_C* A_C = a

A_B = 16/25* A_C

The area of Mirror B depends on the area of Mirror C. Now let's calculate the area of Mirror C!

Area of Mirror C

We are given that the perimeter of Mirror C is 280 centimeters. Since the mirrors are all squares, we know that each mirror has 4 equal sides. This means that we can find the side length Mirror C by dividing the perimeter by 4. Side Length MirrorC= 280/4= 70 Now that we have the side length of Mirror C, we can find its area by recalling the formula for the area of a square. A=s^2 Let's substitute s=70 into the formula to find the area of Mirror C.

A_C=s^2

Substitute

s= 70

A_C=( 70)^2

CalcPow

Calculate power

A_C=4900

The area of Mirror C is equal to 4900 square centimeters. Let's substitute this value into the equation we wrote for A_B to find the area of Mirror B.

A_B = 16/25* A_C

Substitute

C= 4900

A_B = 16/25* ( 4900)

MoveRightFacToNum

a/c* b = a* b/c

A_B = 16(4900)/25

Multiply

Multiply

A_B = 78 400/25

CalcQuot

Calculate quotient

A_B =3136

Area of Mirror A

Finally, let's substitute A_B = 3136 into the equation we wrote for A_A. This will give us the area of Mirror A.

A_A = 25/36* A_B

Substitute

A_B= 3136

A_A = 25/36* ( 3136)

MoveRightFacToNum

a/c* b = a* b/c

A_A = 25(3136)/36

Multiply

Multiply

A_A = 78 400/36

UseCalc

Use a calculator

A_A= 2177.777777 ...

RoundDec

Round to 2 decimal place(s)

A_A=2177.78

The area of Mirror A is about 2177.78 square centimeters.