Volume and Displacement “Satisfaction lies in the effort, not in the attainment. Full effort is full victory.” Mohandas K. Gandhi

Objectives Discover the formula for the volume of a sphere Apply volume formulas to problems involving spheres or hemispheres. Find volumes of irregularly shaped solids through displacement.

What do we know about spheres? Vocabulary: Radius, Center, Great Circle radius center great circle

What is volume? Volume is the measure of the amount of ________ contained in a __________. space solid

Volume of a Sphere Investigation: The Formula for the Volume of a Sphere p542 This investigation demonstrates the relationship between the volume of a hemisphere with radius r and the volume of right cylinder with base radius r and height 2r. r r 2r

Investigation: Steps 1 and 2 After you poured the contents of the hemisphere into the cylinder, what fraction of the cylinder does the hemisphere appear to fill?

Investigation: Step 3 After you filled the hemisphere the second time and pour the contents into the cylinder, what fraction of the cylinder was filled with water from the two hemispheres (or one ___________)? sphere cylinder

Investigation: Step 4 If the radius of the cylinder is r and its height is 2r, then what is the volume of the cylinder in terms of r? V cylinder = Base Area * Height V = (  r 2 )2r V = 2  r 3 r 2r

Investigation: Step 5 The volume of a sphere is the fraction of the cylinder’s volume that was filled by two hemispheres (or one __________). What is the formula for the volume of a sphere? sphere Remember when… we found the area of a sector?

Sphere Volume Conjecture The volume of a sphere with radius r is given by the formula ______.

Hemisphere The volume of a hemisphere with radius r is given by the formula:

Objectives Discover the formula for the volume of a sphere Apply volume formulas to problems involving spheres or hemispheres. Find volumes of irregularly shape solids through displacement.

Example Find the volume of plastic (to the nearest cubic inch) need to make this bowl. The outer-hemisphere diameter is 6.0in and the inner- hemisphere diameter is 5.0in. 6 in 4 in V = 18   =  in 3  40 in 3

Example 6 cm

Objectives Discover the formula for the volume of a sphere Apply volume formulas to problems involving spheres or hemispheres. Find volumes of irregularly shape solids through displacement.

Displacement For a shape for which we have no formula for calculating volume, it is possible to find the volume by submerging the object in a liquid. Think of when you step into a bathtub that is filled to the brim. The volume of the liquid that overflows in each case equals the volume of the solid below the liquid level. This volume is called an object’s ______________. displacement

Example 10 cm 15 cm 2 cm What is the volume of the rock? V = 10 * 15 * 2 = 300 cm 3 10 cm 15 cm

Density An important property of a material is its density. Density is the _______ of matter in a given __________. Formula: mass volume

Example A square-prism container with a base 5cm by 5cm is partially filled with water. You drop a clump of metal that weighs 525 g into the container, and the water level rises 2cm. What is the density of the metal? Assuming the metal is pure, what is the metal? 5 cm 2 cm Silver

Example

Objectives Discover the formula for the volume of a sphere Apply volume formulas to problems involving spheres or hemispheres. Find volumes of irregularly shape solids through displacement.

Closing The formula for the volume of a sphere is ___________. You can find the __________ of an ___________ shaped object by measuring the amount of liquid it __________. Volumes can be used to find the ___________ of objects and thus identify the materials of which they are made. The formula to find density is _____________. volumeirregularly displaces density