Pearson Geometry Common Core, 2011
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Pearson Geometry Common Core, 2011 View details
6. Surface Areas and Volumes of Spheres
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Exercise 50 Page 739

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

Volume: 623Ď€ cubic centimeters
Surface Area: 26Ď€ square centimeters

Practice makes perfect

Let's analyze the given figure.

The solid consists of three smaller ones.

  • A cylinder with a radius of r= 2 cm and a height of h=2.5 cm.
  • Two hemispheres with a radius of 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=Ď€ r^2 h V=1/2*4/3Ď€ r^3
    V_1=Ď€ ( 2)^2( 2.5)= 10Ď€ V_2=1/2*4/3Ď€ ( 2)^3= 16/3Ď€
    The volume of the cylinder is 10Ď€ cubic inches, and the volume of the hemisphere is 163Ď€ cubic inches. Now, let's the volume of the given composite solid, V_\text{solid}.
    V_\text{solid}={\color{#FD9000}{V_1}}+2{\color{#A800DD}{V_2}}
    â–Ľ
    Substitute values and evaluate
    V_\text{solid}={\color{#FD9000}{10\pi}}+2 \left( {\color{#A800DD}{\dfrac{16}{3}\pi}} \right)
    V_\text{solid}=10\pi+2\cdot\dfrac{16\pi}{3}
    V_\text{solid}=10\pi+\dfrac{32\pi}{3}
    V_\text{solid}=\dfrac{3\cdot 10\pi}{3}+\dfrac{32\pi}{3}
    V_\text{solid}=\dfrac{30\pi}{3}+\dfrac{32\pi}{3}
    V_\text{solid}=\dfrac{62\pi}{3}
    V_\text{solid}=\dfrac{62}{3}\pi
    We found that the volume of the composite solid is 623Ď€ cubic centimeters.

    Surface Area

    Now, we will find the surface area of the given composite solid. Notice that it is equal to the surface area of two hemispheres and the lateral 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
    Radius r= 2 r= 2
    Height h= 2.5 -
    Area A=1/2* 4Ď€ r^2 A=2Ď€ r h
    A_1=1/2* 4Ď€ ( 2)^2= 8Ď€ A_2=2Ď€( 2)( 2.5)= 10Ď€
    Now, let's find the surface area of the given solid.
    A_\text{solid}=2{\color{#FF0000}{A_1}}+{\color{#FD9000}{A_2}}
    A_\text{solid}=2\cdot{\color{#FF0000}{8\pi}}+{\color{#FD9000}{10\pi}}
    A_\text{solid}=16\pi+10\pi
    A_\text{solid}=26\pi
    Therefore, the surface area of the composite solid is 26Ď€ square centimeters.