Exercise 8 Page 574

Given the circumference of a circle, we can find its radius. Using this, we can find the diameter and the area of the circle.

Ratio of the Areas: 19
Ratio of the Diameters: 13
Ratio of the Radii: 13

Practice makes perfect

We are told that two circles have circumferences of π and 3π. We are asked to find the ratios of the areas, diameters, and radii of the circles. Let's consider each of these measurements separately, starting with the radii.

Ratio of the Radii

Let's recall the formula for the circumference of a circle. C = 2π r Here, C is the circumference and r is the radius of the circle. Let's find the radius r_1 of the circle with circumference π.

C = 2π r_1

Substitute

C= π

π = 2π r_1

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Solve for r_1

DivEqn

.LHS /2π.=.RHS /2π.

π/2π = r_1

ReduceFrac

a/b=.a /π./.b /π.

1/2 = r_1

RearrangeEqn

Rearrange equation

r_1 = 1/2

The radius r_1 of the first circle is equal to 12. Now let's find the radius r_2 of the second circle. This time, the circumference is 3π.

C = 2π r_2

Substitute

C= 3π

3π = 2π r_2

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Solve for r_2

DivEqn

.LHS /2π.=.RHS /2π.

3π/2π = r_2

ReduceFrac

a/b=.a /π./.b /π.

3/2 = r_2

RearrangeEqn

Rearrange equation

r_2 = 3/2

The radius r_2 of the second circle is equal to 32. Now let's find the ratio of the radius of the first circle to the radius of the second circle.

.r_1 /r_2.

SubstituteII

r_1= 1/2, r_2= 3/2

. 1/2 / 3/2.

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Simplify

DivFracByFracSL

.a/b /c/d.=a/b*d/c

1/2* 2/3

MultFrac

Multiply fractions

1* 2/2*3

CrossCommonFac

Cross out common factors

1* 2/2 * 3

SimpQuot

Simplify quotient

1/3

The ratio of the radius of the first circle to the radius of the second circle is 13.

Ratio of the Diameters

Now we will find the ratio of the diameter of the first circle to the diameter of the second circle. The diameter d of a circle is two times the radius r of the circle. d = 2 r Let's find the diameter of the first circle, which we found has a radius of 12.

d_1 = 2r_1

Substitute

r_1= 1/2

d_1 = 2( 1/2)

DenomMultFracToNumber

2 * a/2= a

d_1 = 1

The diameter d_1 of the first circle is equal to 1. Now let's find the diameter of the second circle. This time, the radius r_2 is equal to 32.

d_2 = 2r_2

Substitute

r_2= 3/2

d_2 = 2( 3/2)

DenomMultFracToNumber

2 * a/2= a

d_2 = 3

The diameter d_2 of the second circle is 3. Now we can find the ratio of the diameter of the first circle to the diameter of the second circle. d_1/d_2 ⇔ 1/3 The ratio of the diameters of the circles is 13.

Ratio of the Areas

Finally, let's find the ratio of the areas of the two circles. Recall the formula for the area of a circle. A = π r^2 Let's find the areas of both circles, starting with the first one. Remember, the radius r_1 is 12.

A_1 = π r_1^2

Substitute

r_1= 1/2

A_1 = π ( 1/2)^2

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Simplify right-hand side

PowQuot

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

A_1 = π 1^2/2^2

CalcPow

Calculate power

A_1 = π 1/4