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=haan601haan601 Chapter 9 Link= = = =7.1 __Surface Area and Volume__= Objectives: For a rectangular prism: S is surface area, l is length, w is width, and h is height S = 2lw + 2lh + 2wh
 * 1) Explore ratios of surface are to volume.
 * 2) Develop the concepts of maximizing volume and minimizing surface area.
 * Surface Area and Volume Formulas**

V is volume and l is length, w is width, and h is height V = lwh

For a cube S is surface area, s is side S = 6s^2

V is volume, s is side V = s^3

If the length of a rectangular prism is 4, the height 6, and the width 2. Using the formulas above, find the surface area and volume.
 * Example**

S = 2(4)(2) + 2(4)(6) + 2(2)(6) S = 16 + 48 + 24 S = 88 V = lwh V = (4)(2)(6) V = 48

=**7.2 __Surface Area and Volume of Prisms__**= Objectives: The surface area S, of a right prism with lateral area L, base area B,perimeter p, and h is S=L+2B or S=hp+ 2b. If two solids have equal hights and the cross selections formed by evry plane parallel to the bases of both solids have equal aareas, then the two soilds have equal volumes. The volume, v, of s prism with hight h and base area B is V=Bh.
 * 1) Define and use a formula for finding the surface area of a right prism.
 * 2) Define and use a formula for finding the volume of a right prism.
 * 3) Use Cavalieri's Principle to develop a formula for the volume of a right or oblique prism.
 * Surface Area of a Right Prism:**
 * Cavalieri's Priniple:**
 * Volume of a prism:**
 * Altitude:** A segment that has endpoints in the planes containing the bases and is perpendicular to both planes.
 * Height:** The lenght of the altitude.

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:**

=7.3 __Surface Area and Volume of Pyramids__=
 * Definitions:**
 * Pyramid**- A polyhedron in which all but one of the polygonal faces intersect at a single point known as the vertex of the pyramid.
 * Base-** The bottom and top of a structure.
 * Lateral face-** The faces of a prism or pyramid that are not bases.
 * Vertex of the pyramid-** The lateral faces are triangles that share a single vertex.
 * Base Edge-** Each lateral faces are triangles that share a single vertex.
 * Lateral edge-** The intersection of two lateral faces of a polyhedron.
 * Altitude-** The perpendicular segment from the vertex to the plane of the base.
 * Height-** The length of an altitude of a polygon.
 * Regular pyramid-** A pyramid whose base is a regular polygon and whose lateral faces are congruent.
 * Slant height-** In a regular pyramid, the length of an altitude of a lateral face.

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__:**

S= L+B or S= 1/2lp+B
 * Surface Area of a regular pyramid:**

V= 1/3Bh
 * Volumes of a Pyramid**

=**7.4 __Surface Area and Volume of Cylinders__**= Objectives:
 * 1) Define and use a formula for the surface area of a right cylinder.
 * 2) Define and use a formula for the volume of a cylinder.

Cylinder-** A solid that consists of a circular region and its translted image on a parallel plane.
 * Definitions:
 * Lateral Surface-** The curved surface of a cylinder or cone.
 * Altitude-** A segment that has endpoint in the planes containing the bases and is perpendicular or both planes.
 * Height-** The lenght of an altitude
 * Axis-** A cylinder is the segment joining the centers of the two bases.
 * Right Cylinder-** A cylinder whose axis is perpendicular to the base.
 * Oblique Cylinder-** A cylinder that is not a right cylinder.

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,

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

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
 * Example:** A penny is a right sylinder with a diameter of 19.05 mm and a thicknoss of 1.55 mm. Ignoring the design,
 * Solution:** The radius of a penny is half of the diameter, of 90525 mm. Use the formula for teh SA of a right cylinder.

=**7.5 __Surface Area and Volume of Cones__**= Objectives:
 * 1) Define and use the formula for the surface area of a cone.
 * 2) Define and use the formula for the volume of a cone.

Cone-** A three dimensional figure that consists of a circular base and a curved lateral surface that connects the base to the base.
 * Definitions:
 * Base-** The circular face of a cone.
 * Lateral surface-** The curved surface of a cylinder that are not bases.
 * Altitude-** A segment from the vertex perpendicular to the plane of the base.
 * Height-** The length of an altitude of a polygon.
 * Right Cone-** A cone in which the altitude intersects the base at its center point.
 * Oblique Cone-** A cone that is not a right cone.
 * Slant Height of a cone-** The length of an altitude of a lateral face.

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

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:**

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:**

=7.6 __Surface Area and Volume of Sphere__= Objectives:
 * 1) Define and use the formula for the surface area of a sphere.
 * 2) Define and use the formula for the volume of a sphere.


 * Sphere-** The set of points in space that are equidistant from a given point know as the center of the sphere.


 * Annulus-** The region between twho circles in a plane that have the same center but different radii.

r-radius S = 4(pi)r² V = 4/3(pi)r³
 * Surface Area and Volume Formulas for a Sphere**
 * Surface Area Formula**
 * Volume Formula**

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:**

=**7.7 __Three-Dimensional Symmetry__**= Objectives
 * 1) Define various transformations in three-dimensional space.
 * 2) Solve problems by using transformations in three-dimensional space.

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//**