koleri

Erin's Chapter 9 = = = = =**7.1** **Surface area and Volume**= Surface area S and volume V of a right rectangular prism with length l width w and height h are: 2lw+2wh+2lh and V= lwh The surface area, S and volume, V of a right rectangular prism are: S=2lw+2wh+2lh and V=lwh The surface area, S and volume, V of a cube with side S are : S=6s^2 and V=s^3
 * objectives:**
 * Explore ratios of surface area to volume
 * Develop the concepts of maximizing volume and minimizing surface area
 * __Surface area and Volume Formulas:__**

[|cube nets and examples]


 * Surface area of an object-**the total area of all the exposed surfaces of the object.

they would want to chose the box which has the lesser surface area, because both of the volumes are equivalent. solution: surface area of the first box: 2(8)(5)+2(4)(5)+2(4)(8)= 184. the second box SA: 2(10)(8) + 2(2)(8)+2(2)(10)= 232. money on the packaging.**
 * Example:** of a cereal company is choosing between two box designs with the densions 8 x 4 x 5, and 2 x 8 x 10,
 * They would want to get the first box because it has the lesser surface area, so they would not have to spend as much

practice surface area and volume [|here] = = = = =7.2 Surface Area and Volume of Prisms= Define and use a formula for finding the surface area of a right prism. Define and se a formula for finding the volume of a right prism. Use Cavalieri's Principle to develop a formula for the volume of a right or oblique prism.
 * Objectives:**
 * Definitions:**
 * Altitude:** A segment that has endpoints in the planes containing the bases and is perpendicular to both planes.
 * Height:** The lenght of the altitude.

The surface area, S of a right prism with lateral area L base area B, perimiter p and height h is : S=L+2B prp S=hp+2B
 * Surface Area of a right prism:**

If two solids have the same heights and cross sections formed by all planes of both solids have the same area, that means they also have the same area.
 * What is Cavalieri's Principle?**

experiment with [|Cavalieri's Principle]!

Given that one gallon ~ .134 cubic feet, how many gallons of water will the aquarium hold? __Volume of the aquarium is found by using the volume formula:__ V=Bh=lwh= (110)(50)(7)=38,500 cubic feet __To approximate the volume in gallons, divide by o.134__ V= 38,500/0.134 ~287,313 gallons __To approximate the weight, multiply by 8.33__ weight~ (287,313)(8.33)~2,393,317 pounds
 * Example:** An aquarium in the shape of a right rectangular prism has dimensionsof 110 x50 x 7 feet.
 * solution:**

Volume Of A Prism: the volume, V and height h and base area B of a prism is
 * V=Bh**

=**7.3** Surface Area and Volume of Pyramids= Objectives: Define and use a formula for the volume of a pyramid.** __Defenitions:__ Pyramid-**polyhedron consistancy of base and three or more lateral edges.** Base- **a polygon** Lateral Face**- triangles that share a single vertex, called vertex of pyramid** Vertex of the pyramid- **lateral faces are triangels that share a single vertex** Base Edge- **each lateral face has one edge in common with the base is called base edge.** Lateral edge- **Intersection of two lateral faces** Altitude- **Altitude is perpendicular segment from vertex to plane of base** Height- **Length of altitude** Regular Pyramid- **Pyramid whose base is a regular polygon whose lateral daces are congruent isosceles triangles** Slant Height- **Length of altitude of lateral face of a regular pyramid**
 * Define and use a formula for the surface area of a regular pyramid.

__Find the surface area of a regular square pyramid whose slant height is__ //__l__// __and whose base edge is__ //__s.__// S= L + B S=4 (1/2sl) + s^2 ~ This can be written as follows: S= 1/2 L (4s) + s^2...because 4s is the perimeter of the base.
 * Example 1.**

[|understanding the pyramid]

__**Surface Area and Volume of Pyramids**____:__

S=L+B of S=1/2lp+B V= 1/3Bh
 * Surface Area of a regular pyramid:**
 * Volume of a Pyramid**



The roof of a gazebo is a regular octagonal pyramid with a base edge of 4 feet and a slant height of 6 feet. Find the area of the roof. L=1/2lp=1/2(6)(8x4)=96 square feet
 * Example:**
 * Solution:** the area of the rood is the lateral area of the pyramid.



=7.4 Surface Area and Volume of Cylinders:= 1.Define and use the formula for the surface area of a rigt cylinder 2. Define and use a formula for the volume of a cylinder
 * Objectives:**

planes
 * Cylinder**: is a solid that consists of a circular region and its translated inage on a arallel plane
 * altitude of a cylinder:** a segment that has endpoints in the planes containing the bases and its perpendicular to both
 * axis of a cylinder:** the segment joining the venters of the two bases
 * oblique cylinder:**The axis of a cylinder is not perpendicular to the bases.
 * right cylinder:** The axis of a cylinder is perpendicular to the bases
 * height of a cylinder:** height of the altitude

radius, r and height h is: S-L+2B o S=2(pi)rh+2(pi)r^2
 * Surface Area of a Right Cylinder:**The surfacea rea, S of a right cylinder with ateral area L, base area B,

Volume of a cylinder: the volume, B of a cylinder wih radius r, height h and base area B is
 * V- Bh or V= (pi)r^2h

Example:** A penny is a right sylinder with a diameter of 19.05 mm and a thicknoss of 1.55 mm. Ignoring the design, estimate the surface area of the penny. S=2(pi)rh+2(pi)r^2 S=2(pi)(9.525)(1.55)+2(pi)(9.525)^2 ~663.46 square mm
 * Solution:** The radius of a penny is half of the diameter, of 90525 mm. Use the formula for teh SA of a right cylinder.



[|cylinder pictures found here]

=7.5 Surface area and volume of cones= Define and use the formula for hte surface area of a cone Define and use the formula for the volume of a cone
 * Objectives:**

percent of the total volume of the original volcano was removed by the eruption? __Find the volume of the original volcano__ V-1/3(pi)r^2h=1/3(5^2)(2)~52.4 cubic miles Find the colume of the destroyed cone V=1/3(pi)r^2h=1/3(1^2)(0.5)~0.52 cubic miles __Find the percent of the original volcano removed by the eruption__ (0.52/52.4)100~1%
 * cone:** three dimensional figure that consists of a circular base and a curved lateral surface that connects the base to a single point not in the plane of the base, called the vertex.
 * altitude of a cone:** the perpendicular segment from the vertex to the plane of the base.
 * solution:**

The surface area, S of a right cone with a latera ara L, base of area B radius r and slan height l is: =L+B or S= (pi)rl+(pi)r^2
 * Surface area of a right cone**
 * S**

The volume, V or a cone with radius r height h and base area b is: V=1/3Bh or V=1/3(pi)r^2h
 * Volume of a cone:**



=7.6 Surface area and volume of spheres= Define and use the formula for the surface area of a sphere Define and use the formula for the volume of a sphere
 * objectives:**


 * sphere:** the set of all points in space that are the same distance, r, from a given point known as the center of the sphere

V=4/3 (pi)r^3 S=4 (pi) r^2
 * Volume of a sphere:** the volume, V of a sphere with radius r is:
 * Surface area of a sphere**: The surface area, S of a sphere with radius r is:

Approcimatel how many cubic feet of hot air can it hold? V=4/3(pi)r^3 =4/3(pi)(27)^3 =4/3(19,683)(pi) =26,244(pi) cubic feet~ 82,488 cubic feet
 * Example:** the enbel[e of a hot-air balloon has a radius of 27 feet when fully inflated.
 * solution:**

=4(pi)(27)^2 =4(729)(pi) =2916(pi)~9160.9 square feet Multiply SA by the cost of each square foot of fabric: 9160.9 x 1.31 = ~12,000$
 * Example two:** the enbelppe of a hot air valloon is 54 feet in diameter when inflated. The cost of the fabric used to make the enbelope is 1.31$ oer square foor. Estimate the total cost of the fabric for the balloon
 * solution:** Estimate the SA: S=4(pi)r^2

[|understanding surface area and volume of spheres]



=7.7 Three Dimensional symmetry=
 * Objectives:**
 * Define various transformations in three-dimensional space
 * Solve problems by using transformations in three-dimensional space




 * A three dimensional figure can be reflected across a plane just like a two dimensional figure can be reflected across a line.**

1. Volume of a Trianular Prism 2. Surface Area of a Triangular Prism 3. Volume of a Pyramid 4. Surface Area of a Pyramid 5. Volume of a Cylinder 6. Surface Area of a Cylinder 7. Volume of a Cone 8. Surface Area of a Cone 9. Volume of a Sphere 10. Surface Area of a Sphere
 * BH**
 * L+2B or HP+2B**
 * (1/3)BH**
 * L+B or (1/2)//l//P+B**
 * BH or (pi)r//²// h**
 * L+2B or 2(pi)rh + 2(pi)r²**
 * (1/3)BH or (1/3)(pi)r//²// h**
 * L+B or (pi)r//l// + (pi)r//²//**
 * (4/3)(pi)r³**
 * 4(pi)r//²//**