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Understanding Volumes and Surface Areas of Spheres and Composite Figures

This guide covers the essential concepts of spheres and composite figures, including how to calculate their surface areas and volumes. It explains the definition of a sphere, provides formulas for surface area (4πr²) and volume (4/3πr³), and illustrates these concepts with examples. Additionally, it includes methods to compute volumes and surface areas of composite solid figures by addressing individual components. Practice problems are provided to reinforce learning, allowing readers to sharpen their skills in solving geometrical problems involving solid figures.

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Understanding Volumes and Surface Areas of Spheres and Composite Figures

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  1. Unit 30 SPHERES AND COMPOSITE FIGURES: VOLUMES, SURFACE AREAS, AND WEIGHTS

  2. SPHERES • A sphere is a solid figure bounded by a curved surface such that every point on the surface is the same distance from a point called the center • A round ball, such as a baseball or basketball, is an example of a sphere

  3. SPHERES The surface area of a sphere equals 4r2 where r is the radius The volume of a sphere equals 4/3 r3 where r is the radius

  4. SPHERE EXAMPLE • Determine the surface area and volume of a basketball that has a 12-cm radius: SA = 4r2 = 4(12 cm)2 = 1809.56 cm2Ans V = 4/3 r3 = 4/3 (12 cm)3 = 7238.23 cm3Ans

  5. VOLUMES AND SURFACE AREAS OF COMPOSITE SOLID FIGURES • To compute volumes of composite solid figures • Determine the volume of each simple solid figure separately • Then add or subtract the individual volumes • To compute the surface areas of composite solid figures • Determine the surface area of each simple solid figure separately • Then add or subtract the individual surface areas

  6. 4" 30" 6" 10" 45" COMPOSITE FIGURE EXAMPLE • Determine the total surface area of the figure below, given that the circle and triangle are holes that were cut out • First determine the area of the rectangle • Then subtract the areas of the circle and triangle

  7. 4" 30" 6" 10" 45" COMPOSITE FIGURE EXAMPLE (Cont) • Determine the total surface area of the figure below, given that the circle and triangle are holes that were cut out • Area of rectangle = 30"  45" = 1350 in2 • Area of triangle = ½(10")(6") = 30 in2 • Area of circle = (2")2 = 12.57 in2 • Total surface area of composite figure: = 1350 in2 – 30 in2 – 12.57 in2 = 1307.43 in2Ans

  8. PRACTICE PROBLEMS • Round all answers to the following problems to two decimal places whenever necessary: • Determine the surface area and volume of a sphere with a radius of 14 mm. • A spherical water tank has a diameter of 8 feet. It is to be repainted, and the total cost of the paint job is $0.28 per square foot. Compute the total cost of repainting the tank.

  9. PRACTICE PROBLEMS (Cont) • Round all answers to the following problems to two decimal places whenever necessary: • How many gallons of water will the tank in problem #2 hold? • Compute the surface area of a ball that has a circumference (great circle) of 25 inches.

  10. PRACTICE PROBLEMS (Cont) • Determine the volume of the bullet shown below:

  11. PROBLEM ANSWER KEY • SA = 2463.01 mm2 V = 11494.04 mm3 • $56.30 • 2010.62 gallons • 198.95 in2 • 11434.35 cm3

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