Exercise 38 Page 732

Practice makes perfect

a We have a disk whose radius is 10 meters. After a 90^(∘) sector is cut away from the disk, a cone is formed by it.

We want to find the circumference of the base of the cone. Notice that the base of the cone is formed by the circumference of the disk.

Let's find the circumference of the disk. To do so we use the following formula. C=2π r We will substitute 10 for r and compute the circumference.

C_d=2π r

Substitute

r= 10

C_d=2π(10)

Multiply

C_d=20π

Note that this is the circumference of the whole disk. However, the given disk is missing a quarter sector. Therefore, to find the circumference of the cone we will multiply 20π by 34. C_c&=20π* 3/4 &=15π Therefore, the circumference of the base of the cone is 15 π meters, or approximately 47.12 meters.

b In this part we will find the area of the base of the cone. Thus, we need the radius of the base of the cone. Remember that the circumference of the base of the cone is 15π meters. With this information and using the formula from Part A, we can find its radius of the base.

C_n=2π r

Substitute

C_n= 15π

15π=2π r

DivEqn

.LHS /2π.=.RHS /2π.

7.5=r

RearrangeEqn

Rearrange equation

r=7.5

From here we can use the following formula to find the area of the base. A=π r^2 Let's substitute 7.5 for r and compute the area.

A=π r^2

Substitute

r= 10

A=π(10)^2

CalcPow

Calculate power

A=π* 100

CommutativePropMult

Commutative Property of Multiplication

A=100π

Therefore, the area of the base of the cone is 100π square meters, or approximately 314.16 square meters.

c To find the volume of the cone we will use the following formula.

V=1/3π r^2 In this formula r is the radius of the base of the cone and h is the height of the cone. This means that we need to find the height of the cone. Notice the slant height of the cone is the radius of the disk.