Exercise 51 Page 739

Let's analyze the given figure.

The solid is a cylinder with a hemisphere cut out.

• The cylinder has radius r= 42= 2, and the height h=2.5 cm.
• The hemisphere has radius r= 2 cm.

  We are asked to find the volume and the 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= 2	r= 2
  Height	h= 2.5	-
  Volume	V=π r^2 h	V=1/2*4/3π r^3
  	V_1=π ( 2)^2( 2.5)= 10π	V_2=1/2*4/3π ( 2)^3= 16/3π

  The volume of the cylinder is 10π cubic inches, and the volume of the hemisphere is 163π cubic inches. Now, let's find the difference of these volumes to get the volume of the given composite solid, V_\text{solid}.
  V_\text{solid}={\color{#FD9000}{V_1}}-\textcolor{darkviolet}{V_2}

  ▼

  Substitute values and evaluate

  SubstituteII

  V_1= 10π, V_2= 16/3π

  V_\text{solid}={\color{#FD9000}{10\pi}}-{\color{#A800DD}{\dfrac{16}{3}\pi}}

  MoveRightFacToNum

  a/c* b = a* b/c

  V_\text{solid}=10\pi-\dfrac{16\pi}{3}

  NumberToFrac

  a = 3* a/3

  V_\text{solid}=\dfrac{3\cdot 10\pi}{3}-\dfrac{16\pi}{3}

  Multiply

  Multiply

  V_\text{solid}=\dfrac{30\pi}{3}-\dfrac{16\pi}{3}

  SubFrac

  Subtract fractions

  V_\text{solid}=\dfrac{14\pi}{3}

  MovePartNumRight

  a* b/c=a/c* b

  V_\text{solid}=\dfrac{14}{3}\pi
  Finally, we find that the volume of the composite solid is 143π cubic centimeters.

  Surface Area

  Now, we will find the surface area of the given composite solid. Notice that it's 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= 2	r= 2	r= 2
  Height	h= 2.5	-	-
  Area	A=1/2* 4π r^2	A=2π r h	A=π r^2
  	A_1=1/2* 4π ( 2)^2= 8π	A_2=2π( 2)( 2.5)= 10π	A_3=π ( 2)^2= 4π

  Now, let's add the area to find the surface area of the composite solid, A_\text{solid}.
  A_\text{solid}={\color{#FF0000}{A_1}}+{\color{#FD9000}{A_2}}+{\color{#A800DD}{A_3}}

  SubstituteValues

  Substitute values

  A_\text{solid}={\color{#FF0000}{8\pi}}+{\color{#FD9000}{10\pi}}+{\color{#A800DD}{4\pi}}

  AddTerms

  Add terms

  A_\text{solid}=22\pi
  Therefore, the surface area of the composite solid is 22π square centimeters.