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McGraw Hill Integrated II, 2012 View details

6. Surface Areas and Volumes of Spheres

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Exercise 29 Page 853

Use formula for the volume of a sphere and for the volume of a cylinder.

Volume: About 385.4 cubic inches
Surface Area: About 276.5 square inches

Practice makes perfect

Let's analyze the given composite solid.

The solid consists of two smaller ones.

A cylinder with a radius of r= 4 inches, and the height of h=5 inches.
A hemisphere with a radius of r= 4 inches.

We are asked to find the volume and surface area of the above solid. First, let's find the volume.

Volume

Let's use the formula for the volume of a sphere and for the volume of a cylinder.

Solid	Cylinder	Hemisphere
Radius	r= 4	r= 4
Height	h= 5	-
Volume	V=π r^2 h	V=1/2*4/3π r^3
Volume	V_1=π ( 4)^2( 5)=80π	V_2=1/2*4/3π ( 4)^3=128/3π

The volume of the cylinder is 80π cubic inches and the volume of the hemisphere is 1283π cubic inches. Now, let's add these volumes to get the volume of the given composite solid, V_\text{solid}. We will also round the answer to the nearest tenth.

V_\text{solid}=\textcolor{darkorange}{V_1}+\textcolor{darkviolet}{V_2}

▼

Substitute values and evaluate

SubstituteII

V_1= 80π, V_2= 128/3π

V_\text{solid}=\textcolor{darkorange}{80\pi}+\textcolor{darkviolet}{\dfrac{128}{3}\pi}

MoveRightFacToNum

a/c* b = a* b/c

V_\text{solid}=80\pi+\dfrac{128\pi}{3}

NumberToFrac

a = 3* a/3

V_\text{solid}=\dfrac{3\cdot 80\pi}{3}+\dfrac{128\pi}{3}

Multiply

V_\text{solid}=\dfrac{240\pi}{3}+\dfrac{128\pi}{3}

AddFrac

Add fractions

V_\text{solid}=\dfrac{368\pi}{3}

UseCalc

Use a calculator

V_\text{solid}=385.3686988\ldots

RoundDec

Round to 1 decimal place(s)

V_\text{solid}\approx 385.4

We found that the volume of the composite solid is about 385.4 cubic inches.

Surface Area

Now,we will find the surface area of the given composite solid. Notice that it is equal to the surface area of the hemisphere, the lateral area of the cylinder, and one base area of the cylinder. Let's use the formulas for the surface area of a sphere and for the surface area of a cylinder.

Surface	Hempishere	Lateral Area of Cylinder	Base Area of Cylinder
Radius	r= 4	r= 4	r= 4
Height	h= 5	-	-
Area	A=1/2* 4π r^2	A=2π r h	A=π r^2
Area	A_1=1/2* 4π ( 4)^2= 32π	A_2=2π( 4)( 5)=40π	A_3=π ( 4)^2=16π

Now, let's add the area to find the surface area of the composite solid, A_\text{solid}. Then we will round the answer to the nearest tenth.

A_\text{solid}={\color{#FF0000}{A_1}}+\textcolor{darkorange}{A_2}+\textcolor{darkviolet}{A_3}

▼

Substitute values and evaluate

SubstituteValues

Substitute values

A_\text{solid}={\color{#FF0000}{32\pi}}+\textcolor{darkorange}{40\pi}+\textcolor{darkviolet}{16\pi}

AddTerms

Add terms

A_\text{solid}=88\pi

UseCalc

Use a calculator

A_\text{solid}=276.460135\ldots

RoundDec

Round to 1 decimal place(s)

A_\text{solid}\approx 276.5

Therefore, the surface area of the composite solid is about 276.5 square inches.

Subchapter links

Exercises p.852-855

Exercise 29 Page 853

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Volume

Surface Area