What is the Volume, in Cubic Inches, of a Basketball with a Diameter of 9 Inches

Spherical and Cubic Volume Applications Warm-Up ? How can you apply the formulas for volume of a cube and a sphere to solve problems? Lesson Question Lesson Goals Apply the formulas for volume of a sphere and a cube. Identify perfect cubes. Understand cube roots W 2K Determine side length and . radius measures. Words to Know Write the letter of the definition next to the matching word as you work through the lesson. You may use the glossary to help you. C approximate A. set of all points in space a given distance from a fixed point B perfect cube B. a number that is the result of cubing a natural number D radius A sphere D. a segment that extends from the center of a circle to any point on the circle E volume E. the measure of the amount of space occupied by a three-dimensional solid object C. near or very close to the actual 1 Warm-Up Spherical and Cubic Volume Applications Volume of a Sphere The formula for the volume , V, of a sphere is given by V = 4 3 πr . 3 • What is the volume of a sphere with a diameter of 6 mm? r 2 If the diameter is 6, cut in half, we get d= 6 mm 3 4 3 πr 3 4 V = π(3)3 3 4 V = π(27) 3 V= V= (4)( 36π V= 2 9 )π mm3 for r. Instruction Spherical and Cubic Volume Applications Slide 2 Calculate the Volume What is the volume of this cube with a side length of 5 centimeters? • Cube V = s3 = s ⋅ s ⋅ s V = s3 V= 53 = 5⋅5⋅5 = 5 cm Find the Side Length EXAMPLE What is the side length of a cube with a volume of 8 cm3? • Cube V = s3 3 V = 8 cm3 2 3 8 8 = s3 = 3 s3 cm = s 125 cm3 Instruction Spherical and Cubic Volume Applications Slide 4 The Root Versus Dividing MISCONCEPTIONS What is the side length of a cube with a volume of 27 mm3? Taking the • 3 cube root Dividing 27 = 3 because 33 = 3 ⋅ 3 ⋅ 3 = 27 • 3 27 = 9 because 27 ÷ 3 = 9 9 mm 3 mm 3 mm 3 mm V = 33 = 3 ⋅ 3 ⋅ 3 = 9 mm 27 9 mm V = 93 = 9 ⋅ 9 ⋅ 9 = 729 So this method of dividing is incorrect. Correct method. 4 Instruction Spherical and Cubic Volume Applications Slide 6 Perfect Cubes versus Not-Perfect Cubes • Perfect cubes • Not-perfect cubes • 4 units3 • 1 unit3 1⋅1 ⋅1 = 1 Why is it not a perfect cube? It falls between two 13 < 4 < 23 1<4<8 • 64 units3 4⋅4⋅4 = 64 8⋅8⋅8 = 512 perfect cubes • 12 units3 23 < 12 < 33 • 512 units3 8 < 12 < 27 • 25 units3 23 • 729 units3 9 ⋅ 9 ⋅ 9 = 729 < 25 < 33 8 < 25 < 27 • 600 units3 83 < 600 < 93 512 < 600 < 5 729 . Spherical and Cubic Volume Applications Instruction Slide 9 Find the Radius of a Sphere EXAMPLE The volume of a sphere is 36π cubic centimeters. What is the radius? • Sphere V = 4 3 πr 3 4 3 πr 3 4 = πr 3 3 V= 36π  3  3  4 3 π = (36)  4   4   3  πr V = 36π cm3 27 π π 3 πr 3 = π 27 = r3 27 = 3 r 3=r r= 6 3 cm Instruction Spherical and Cubic Volume Applications Slide 12 2 Volume of a Basketball REAL-WORLD CONNECTION A basketball has a radius of 5 inches. Approximately how much air does it take to fill the basketball? • Sphere V = 4 3 πr 3 4 3 πr 3 4 V = π(5)3 3 V= V= V≈ 7 125    523.6 4 π 3  in.3 Instruction Spherical and Cubic Volume Applications Slide 14 Find the Diameter of a Real-World Sphere Marcel received a globe as a gift. On the box, it states that it has a volume of 972π cubic inches. He wants to find the diameter measure to make sure that it will fit in his shelf space before he opens the box. What is the diameter of Marcel’s globe? 4 V = πr 3 3 972π = 4 πr 3 3 3 4   3 π = 972   4   ( ) 729π 729 = r3 729 = 3 r 3 9 =r r = 9 because 93 = 729 d= 4  πr 3 3  πr 3 = π π 3    9⋅ The diameter of Marcel’s globe is 8 2 18 = 18 in. inches. Summary ? Lesson Question Spherical and Cubic Volume Applications How can you apply the formulas for volume of a cube and a sphere to solve problems? Answer The formulas for volume of a cube and a sphere can be applied to solve problems involving side lengths and radius when finding cube roots. Use this space to write any questions or thoughts about this lesson. 9