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Big Ideas Math Geometry, 2014

BI

Big Ideas Math Geometry, 2014 View details

8. Surface Areas and Volumes of Spheres

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Exercise 47 Page 654

Make a right triangle using the cone's height, the cone's side, and the sphere's radius.

Surface Area: 36π≈ 113.10 inches^2
Volume: 24π ≈ 75.40 inches^3

Practice makes perfect

We have a symmetric cone with a height of 6 inches. Inscribed in this cone is a sphere with a radius of 2 inches. Let's begin by making a diagram of the situation.

A sphere inscribed in a right cone

We will first find the radius of the base area of the cone and the cone's slant height. After that we will find its surface area and its volume. Let's do it!

Radius of the Base

We want to find the volume and the surface area of the cone. By making use of the height of the cone, the radius of the sphere, and the side of the cone, we can form a right triangle.

A sphere inscribed in a right cone. A right triangle with one side being the circle's radius highlighted

Using the sine ratio we can find the measure of the angle α. sin(θ)=Opp./Hyp. ⇒ sin( α)=2/4 Let's solve the equation.

sin(α)=2/4

▼

Solve for α

a/b=.a /2./.b /2.

sin(α)=1/2

sin^(-1)(LHS) = sin^(-1)(RHS)

α=sin^(- 1)(1/2)

300sin^(-1)(0)=0^(∘) 3030sin^(-1)(1/2)=30^(∘) 3045sin^(-1)(sqrt(2)/2)=45^(∘) 3060sin^(-1)(sqrt(3)/2)=60^(∘) 3090sin^(-1)(1)=90^(∘) 30-30sin^(-1)(- 1/2)=- 30^(∘) 30-45sin^(-1)(- sqrt(2)/2)=- 45^(∘) 30-60sin^(-1)(- sqrt(3)/2)=- 60^(∘) 30-90sin^(-1)(- 1)=- 90^(∘)

α= 30^(∘)

We can now draw a radius of the base of the cone.

A sphere inscribed in a right cone. A right triangle including the radius of the base of the cone is marked

The radius, the height, and the side of the cone form a right triangle. We can use the tangent ratio to find the length of the radius. tan(θ)=Opp./Adj. ⇒ tan( 30)=r/6 Let's solve this equation.

tan(30)=r/6

▼

Solve for r

300tan(0^(∘))=0 3030tan(30^(∘))=sqrt(3)/3 3045tan(45^(∘))=1 3060tan(60^(∘))=sqrt(3) 3090tan(90^(∘)) undefined 30120tan(120^(∘))=- sqrt(3) 30135tan(135^(∘))=- 1 30150tan(150^(∘))=- sqrt(3)/3 30180tan(180^(∘))=0 30270tan(270^(∘)) odef. 30360tan(360^(∘))=0

sqrt(3)/3=r/6

LHS * 6=RHS* 6

(6)sqrt(3)/3=r

MoveLeftFacToNum

a*b/c= a* b/c

6sqrt(3)/3=r

a/b=.a /3./.b /3.

2sqrt(3)=r

Rearrange equation

r= 2sqrt(3)

Slant Height

Now we need to find the slant height of the cone, l.

A sphere inscribed in a right cone. A right triangle including the radius of the base of the cone is marked

We can use the Pythagorean Theorem to find l. a^2+b^2=c^2 Let's substitute the corresponding values into the equation and solve for l.

a^2+b^2=c^2

SubstituteValues

Substitute values

6^2 +( 2sqrt(3))^2= l^2

▼

Solve for l

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

6^2+2^2(sqrt(3))^2=l^2

Calculate power

36+4(sqrt(3))^2=l^2

( sqrt(a) )^2 = a

36+4(3)=l^2

Multiply

36+12=l^2

Add terms

48=l^2

Rearrange equation

l^2=48

sqrt(LHS)=sqrt(RHS)

l= sqrt(48)

Surface Area

Recall the formula for the surface area of a cone. S=π r^2 + π r l Let's substitute 2sqrt(3) for r and sqrt(48) for l to find the surface area.

S=π r^2 + π r l

r= 2sqrt(3), l= sqrt(48)

S=π ( 2sqrt(3))^2 + π ( 2sqrt(3)) ( sqrt(48))

▼

Simplify right-hand side

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

S=π (2^2)(sqrt(3))^2 + π (2sqrt(3)) (sqrt(48))

Calculate power

S=π (4)(sqrt(3))^2 + π (2sqrt(3)) (sqrt(48))

( sqrt(a) )^2 = a

S=π (4)(3) + π (2sqrt(3)) (sqrt(48))

Multiply

S=π (12) + π (24)

CommutativePropMult

Commutative Property of Multiplication

S=12π + 24π

Add terms

S=36π

The surface area of the cone is 36π≈ 113.10 inches^2.

Volume

The volume of a cone is one third times the area of the base and the height. V=1/3Bh Our cone has a circular base and its area is the product of π and the radius squared. V=1/3Bh ⇒ V=1/3π r^2 h Let's use this formula to find the volume of the cone.

V=1/3π r^2 h

r= 2sqrt(3), h= 6

V=1/3π ( 2sqrt(3))^2 ( 6)

▼

Simplify right-hand side

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

V=1/3π (2^2)(sqrt(3))^2 (6)

Calculate power

V=1/3π (4)(sqrt(3))^2 (6)

( sqrt(a) )^2 = a

V=1/3π (4)(3) (6)

Multiply

V=1/3π (72)

CommutativePropMult

Commutative Property of Multiplication

V=1/3(72)π

MoveRightFacToNumOne

1/b* a = a/b

V=72π/3

a/b=.a /3./.b /3.

V=24π

The cone's volume is 24π ≈ 75.40 inches^3.

Subchapter links

Communicate Your Answer p.647

Monitoring Progress p.649-651