torgab

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__Chapter 7 Objectives__
•explore ratios of surface area to volume •Develope the concepts of maximizing volume and minimizing sufrace area

__7.1__ =__Surface Area and Volume__= The surface area of an object is the total area of all the exposed surface of the object. = =

__Surface Area abd Volume Formulas__
-S=2lw+2wh+2lh and V = lwh -S=6s² and V=s³ =__**Example**__= the I pod company is choosing between ipod boxes one box is! which box has a greater surface area? H=8in W=4in L=5in

Related Sites Explaining Surface Area and Volume: Formulashttp://math2.org/math/geometry/areasvols.htm Math Examples Relating to Surface area. http://www.321know.com/geo79_x9.htm Math Examples Relating to Volume. http://www.321know.com/geo79_x7.htm

H=10in W=2in L8in//
 * 2

=__Solution__= both boxes have a volume of 160 cubic inches Box B has a greater surface area.

=7.2= =__Prisms__=

Definitions:
__Altitude-__ a segment with endpoints in the planes containing the bases and that is perpendicular to both of the planes. __Height-__ the lengh S=surface area L=lateral area B=base area P=perimeter H=height S=L+2B or S=hp+2//S=L+2B or S=hp+2
 * __Surface area of a right prism__**

Helpful Websites for Practice http://argyll.epsb.ca/jreed/math9/strand3/3107.htm////////

__Cavalieri's Principle__
In order fot two solids to have equal volumes they have to have equal height and the cross sections formed by every plane parallel to the bases have to have equal areas The Volume of a Prism V=volume h=height B=base area V=Bh =__7.3__= =__Pyramids__=

Definitions:
__Pyramid-__In a polyhedron all of the faces but one of the faces intersects at a single point known as the vertex of the pyramid. __Base-__The face of a polygon that is opposite of the vertex __Lateral face-__faces of a pyramid or prism that are not the bases __Base edge-__Its part of the base of a pyramid that is an edge;each of the lateral faces has only one edge in common with the base __Lateral edge-__In a polyhedron, its the intersection of two lateral faces __Altitude-__segment from the vertex that ia perpendicular to the plane on the base __Height-__A polygons length of an altitude __Regular pyramid-__A pyramid with a base that is a regular polygon and that that lateral faces are congruent isoseles triangles __Slant height-__A regular pyramid, the length of an altitude of a lateral face

__Surface area of a regular pyramid__
L=lateral face B=base area P=perimeter of the base l=slant height

S=L+B or S=1/2lp+B

__Volume of a pyramid__
V=volume H=height B=base area

V=1/3Bh = = =__7.4__= =__Cylinders__=

Definitions:
__Cylinder-__a soid that has a circular region and its translated image in a parallel plane with a lateral surface connecting the circles __Lateral surface-__Its the curved surface on a cone or cylinder __Bases-__faces formed by the circular region and its translated image __Altitude-__a segment thats from the vertex thats perpendicular to the plane of the base __Height-__The length of an altitude of a polygon __Axis-__a segment that joins the centers of the two bases __Right cylinder-__A cylinder that has its axis perpendicular to the bases __Oblique cylinder-__its a cylinder that is not a right cylinder

__Surface Area of a Right Cylinder__
S=surface area L=lateral area B=base area r=radius h=height

S=L+2B or S=2pierh+2pier²

__Volume of a Cylinder__
V=volume r=radius h=height B=base area

V=Bh or V=pier²h

=__7.5__= =__Cones__=

Definitions:
__Cone-__a three dimensional figure that has a circlular base and a lateral face __Base-__the circular face of a cone Lateral surface-The curved part of a cone or cylinder __Altitude-__A segment from the vertex thats perpendicular to the plane of the base __Height-__The lenght of the altitude __Right cone-__A cone, were the altitude intersects the base at its center point __Oblique cone-__A cone thats not a right cone __Slant height of a cone-__the radius of the sector

__Surface area of a right cone__
S=surface area L=lateral face B=base of area r=radius l=slant height

S=L+B or S=pierl+pier²

__Volume of a cone__
V=volume r=radius h=height B=base area

V=1/3Bh or V=1/3pier²h =__7.6__= =__Spheres__=

Definitions: Sphere-is the set of all points in space that are the same distance from a given point known as the center of the sphere Annulus-A ring,a shaped figure in the cylinder

V=volume r=radius
 * __Volume of a Sphere__**

V=4/3pier³

S=surface area r=radius
 * __Surface Area of a Sphere__**

S=4pier²

1. volume of triangular prism: For a class project we are given a triangular prism with a width of 7, a lenght of 8 and a height of 5. We need to find the volume of this prism. V=(1/2)whl V=(1/2)7*8*5 V=140

2. surface area of triangular prism: Using the same triangular prism as above figure out the surface area of it. S= bh + (s1 + s2 + s3)H S= 56*5 +(30)5 S=430 3. volume of pyramid: A pyramid in Egypt is being built. The designers have two ideas in mind. They want there pyramid to have the greatest volume possible. The first pyrmid has a height of 100ft and a base area of 150ft. The second pyramid has a height of 160ft and a base area of 105ft. What pyramid should the designers build that have the greatest volume? first pyramid: V=(1/3)Bh V=(1/3)150*100 V=5000 second pyramid= V=(1/3)Bh V=(1/3)105*160 V=5600 They should build the second pyramid.

4. surface area of pyramid: The same designers that were building the pyramid before want to find out the surface area of the pyramid there going to build. The pyramid has a base area of 105ft, a length of 10 and a slant height of 100ft. Find the surface area. S=(1/2)lp+B S=(1/2)10*100+105 S=605

5. volume of cylinder: A juice company is coming out with a new juice and they want to design a new container to put it in. They have come up with two designs and want to have the must volume in the container. The first container has a diameter of 6 and a height of 8. The second design has a diameter of 4.5 and a height of 10. What container should the company use? first container: V= (phi)r^2h V=(phi)3^2*8 V=226.19 second container: V=(phi)r^2h V=(phi)2.25^210 V=159.04 They should choose the first container.

6. surface area of cylinder: The same juice company as before is now trying to figure out what the surface area is of the choosen container so they can buy a wrapping for it. If the container has a diameter of 6 and a height of 8, and the cost for a square inch of wrapping is .25 cents. What is it gonna cost to wrap the container? S= 2(phi)r 2+2(phi)rh S=(phi)3^2+2(phi)3*8 S=179.07 179.07*.25 = $44.77

7. volume of cone: An ice cream shop is deciding what cone they want to have a special on. They want the cone to have the least amount of volume. They bought two kinds of cones from the cone shop, the first cone has a height of 7 and a diameter of 3, the second cone has a height of 6.5 and a diameter of 4. What cone has the least volume? first cone: V= (1/3)(phi) r 2 h V=(1/3)(phi)1.5^2*7 V=16.49 second cone: V= (1/3)(phi) r^2 h V= (1/3)(phi)2^2*6.5 V=27.23 The should put the second cone on special.

8. surface are of cone: The same ice cream shop is having another special but this time they want to have the least amount of surface area on a cone to use. Using the same cones as above which cone has the lesser amount of cone? Each cone has a slant height of 6. first cone: S= (phi)rl + (phi)r^2 S= (phi) 1.5*6 + (phi)1.5^2 S=35.34 second cone: S= (phi)rl + (phi)r^2 S= (phi) 2*6 + (phi)2^2 S=50.26 The should choose the first cone.

9. volume of sphere: For my birthday I want to have the biggest red ball ever. In a magazine there is two different balls i want, there are no pictures but only the diameter of each ball. The first ball has a diameter of 15 and the second ball has a diameter of 15.5. Which ball should I choose? first ball: V= (4/3) (phi) r^3 V=(4/3) (phi) 7.5^3 V= 1767.14 second ball: V= (4/3) (phi) r^3 V= (4/3) (phi) 7.75^3 V= 1949.82 I should choose the second one.

10. surface area sphere: My Grampa is gonna get me the second ball but he has decided to wrap it. He wants to know how much surface area this ball has so she knows how much wrapping paper to get. Figure out the surface area. S= 4(phi)r^2 S=4(phi)7.75^2 S=754.77