Download - Assignment
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Assignment• P. 842-5: 2, 3-11
odd, 12-20 even, 21-23, 28, 33-38
• P. 850-3: 1, 2, 3-21 odd, 24, 30
• Solids of Revolution Worksheet
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Warm-UpDraw and name the 3-D solid of revolution
formed by revolving the given 2-D shape around the x-axis.
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Warm-UpDraw and name the 3-D solid of revolution
formed by revolving the given 2-D shape around the x-axis.
Sphere Hemisphere Torus
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12.6-12.7: Volume and Surface Area of Spheres and Similar Solids
Objectives:1. To derive and use the formulas for the
volume and surface area of a sphere2. To find the surface area and volume of
similar solids
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SphereA sphere is the set
of all points in space at a fixed distance from a given point.
• Radius = fixed distance
• Center = given point
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Exercise 1What is the result of
cutting a sphere with a plane that intersects the center of the sphere?
What 2-D shape is projected onto the plane?
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HemisphereA hemisphere is
half a sphere. The circle on the base of a hemisphere is a great circle.
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Investigation 1In this Investigation, we
will discover the formula for the volume of a sphere. To do this we need to relate the sphere to a very particular cylinder.
Sphere CylinderRadius = r Radius = r
Height = 2r
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Investigation 1In this Investigation, we
will discover the formula for the volume of a sphere. To do this we need to relate the sphere to a very particular cylinder.
Notice that this is the largest possible sphere that could fill the cylinder. This sphere is inscribed within the cylinder.
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Investigation 1Step 1: Rather than use the
sphere, we’ll use the hemisphere with the same radius, since it will be easier to fill. So…fill the hemisphere.
Step 2: Pour the contents of the hemisphere into the cylinder. How full is it?
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Investigation 1Step 3: Repeat steps 1
and 2. How full is the cylinder?
Step 4: Repeat step 3. How full is the cylinder? What does this tell you about the volume of the sphere?
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Archimedes TombArchimedes was the first
to discover that the volume of a sphere is 2/3 the volume of the cylinder that circumscribes it. He considered this to be his greatest mathematical achievement.
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Exercise 2 Derive a formula for the
volume of a sphere.
Sphere Cylinder23
V V
223
r h
22 23
r r
343
r
h = 2r
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Exercise 3Derive a formula
for the volume of a hemisphere.
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Exercise 4What is the extended ratio of the volume of
the cone to the sphere to the cylinder?
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Volume of Spheres and Hemispheres
Volume of a Sphere
• r = radius of the sphere
Volume of a Hemisphere
• r = radius of the hemisphere
343V r 32
3V r
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Find the volume of each solid using the given measure.
1. d = 18.5 inches 2. C = 24,900 miles
Exercise 5
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Find the volume of each solid using the given measures.
1. V = 2. V =
Exercise 6
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Investigation 2Now we’ll find a
formula for the surface area of a sphere. To do this, perhaps we should use a net…
Or perhaps we’ll look at it another way.
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Investigation 2Think of a sphere as
being constructed by a whole bunch of pyramids—I mean bunch of them. The height of each pyramid would be the radius of the sphere.n = a whole bunchh = radius of the sphere B
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Investigation 2Let’s also say that each
of these pyramids is congruent and has a base area of B.
Thus, the surface area of the sphere is:
1 2 3 nS B B B B (Not a very useful formula)
B
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Investigation 2Furthermore, the volume
of the sphere should be the sum of the volumes of the pyramids.
1 1 1 11 2 33 3 3 3 nV B h B h B h B h
11 2 33 nV h B B B B
11 2 33 nV r B B B B
13V r S B
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Exercise 7Use the two formulas
below to derive a formula for the surface area of a sphere.
13V r S
343V r
B
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Exercise 8Explain how the
unwrapped baseball illustrates the formula for the surface area of a sphere.
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Exercise 9 Derive a formula for the total surface area of
a hemisphere.
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SA of Spheres and Hemispheres
Surface Area of a Sphere
• r = radius of the sphere
Surface Area of a Hemisphere
• r = radius of the hemisphere
24S r 23S r
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Find the surface area of each solid using the given measure.
1. d = 18.5 inches 2. C = 24,900 miles
Exercise 10
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Similar SolidsAny two solids are similar solids if they are of the
same type such that any corresponding linear measures (height, radius, etc.) have equal ratios.– Ratio = scale factor
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Exercise 11Explain why any two
cubes are similar.
2"
4"
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Exercise 12Find the volume of a
cube with a side length of 2 inches.
Now find the volume of a cube with a side length of 4 inches.
How do the volumes compare?
2"
4"
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Exercise 12Find the volume of a
cube with a side length of 2 inches.
Now find the volume of a cube with a side length of 4 inches.
How do the volumes compare?
2"
4"2" 2"
2" 2"
2" 2"2" 2"
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Volumes of Similar FiguresIf two solids have a
scale factor of a:b, then the corresponding volumes have a ratio of a3:b3.
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Similarity Relationships
Perimeter Linear Units a:b
Area Square Units a2:b2
Volume Cubic Units a3:b3
For two shapes with a scale factor of a:b, each of the following relationships will be true.
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Exercise 13A breakfast-cereal manufacturer
is using a scale factor of 5/2 to increase the size of one of its cereal boxes. If the volume of the original cereal box was 240 in.3, what is the volume of the enlarged box?
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Exercise 14Pyramids P and Q are similar. Find the scale factor
of pyramid P to pyramid Q.
V = 1000 in3 V = 216 in3
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Assignment• P. 842-5: 2, 3-11
odd, 12-20 even, 21-23, 28, 33-38
• P. 850-3: 1, 2, 3-21 odd, 24, 30
• Solids of Revolution Worksheet