wriang

wran38 <-- chapter 9 link

Objectives:** explore ratios of surface area to volume. develop the concepts of maximizing surface area.
 * __7.1__

surface area and volume of a rectangular prism: S=2lw+2wh+2lh V=lwh
 * blue box:**

Surface area and volume of a rectangular prism: S=6s^2 V=s^3

A rice company is choosing between 2 box designs with the diimensions of the first box 8in. by 4in. by 5in. and the 2nd box being 2in. by 8in. by 10in. which design has the greater surface area and requires more material for the same volume? both have a volume of 160in. The surface area of a box 1 is 2(8)(5)+2(4)(5)+2(4)(8)=184in^2. The surface area of the 2nd box is 2(10)(8)+2(2)(8)+2(2)(10)=232in^2. The 2nd box has the greater surface area.
 * Example:**

http://www.flickr.com/photos/elmada/254689286/

Objectives:** Define and use a formula for finding the surface area of a right prism. Define and use 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.
 * __7.2__


 * Altitude:** a segment that has endpoints in the planes containg the bases and that is perpendicular to both planes.
 * Height:** the length of an altitude.

Surface area of a right prism: S=L+2B or S=hp+2B
 * blue boxes:**

Cavalieri's principle: If two solids have equal heights and the cross sections formed by every plnae parallel to the bases of both solids have equal areas, then the two solids have equal volumes.

Volume of a prism: V=Bh A fish tank has the shape of a right rectangular prism. It has the dimensions of 3ft by 2ft by 1ft. What is the volume of the fish tank? V=Bh so V=(3)(2) V=6
 * Example:**

http://www.flickr.com/photos/hi-phi/59870693/

Objectives:** Define and use a formula for the surface area of a regular pyramid. Define and use a formula for the volume of a pyramid.
 * __7.3__


 * Pyramid:** A polyhedron consisting of a base, and three or more lateral faces.
 * Base:** The polygonal face that is opposite the vertex.
 * Lateral Faces:** The faces of a prism or pyramid that are not bases.
 * Base Edge:** An edge that is part of the base of a pyramid, each lateral face has one edge in common with the base.
 * Lateral Edge:** The intersection of two lateral faces of a polyhedron.
 * Altitude:** A segment from the vertex perpendicular to the line containing the opposite side.
 * Height:** The length of an altitude of a polygon.
 * Regualr Pyramid:** A pyramid whose base is a regular polygon and whose lateral faces are congruent isosceles triangels.
 * [|Slant Height:]** In a regular pyramid, the length of an altitude of a lateral face.

surface area of a regular pyramid: S=L+B or S=(1/2)lp+B
 * Blue Box:**

Volume of a pyramid: V=(1/3)Bh

The roof of a porch 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. If roofing material costs $4.50 per square feet, find the cost of covering the roof with the material. L=(1/2)lp=(1/2)(6)(8*4)= 96 square feet 96 square feet * $4.50 per square foot = $432.00
 * Examples:**

The pyramid is a regular square pyramid with a base edge of 600 feet and an orginal height of 400 feet. The rock used to construct the pyramid weights about 200 pounds per cubic foot. Estimate the weight of the pyramid. V=(1/3)Bh V=(1/3)(600^2)(400) V=48,000,000 The weight in pounds is 48,000,000 cubic feet * 200 pounds per cubic foot = 9,600,000,000 pounds.

http://www.flickr.com/photos/ny_doll/295838684/

Objectives:** Define and use a formula for the surface area of a right cylinder. Define and use a formula for the volume of a cylinder.
 * __7.4__


 * Cylinder:** a solid that consists of a circular region and its translated image on a parallel plane
 * Lateral Surface:** The curved surface of a cylinder or cone.
 * Bases:** The faces formed by the circular region and it's translated image.
 * Altitude:** A segment that has endpoints in the plane containing the bases and is perpendicular to both planes.
 * Height:** The length of an altitude of a polygon.
 * Axis:** The segment joining the centers of the 2 bases.
 * Right Cylinder:** A cylinder whose axis is perpendicular to the bases.
 * [|Oblique Cylinder:]** A cylinder that is not a right cylinder.

Surface Area of a Right Cylinder: S=L+2B or S=2(phi)rh+2(phi)r^3
 * Blue Boxes:**

Volume of a Cylinder: V=Bh or V=(phi)r^3h

A dime is a right cylinder with a diameter of 16.00 millimeters and a thickness of 1.55 millimeters. Estimate the surface area of a penny. The radius of a dime is half the diameter, so the radius is 8.00 millimeters. S=2(phi)rh+2(phi)r^2 S=2(phi)(8.00)(1.55)+2(phi)(8.00)^2 = 480.03 square millimeters.
 * Examples:**

A tank has a height of 20 feet and a diameter of 14 feet. What is the volume of this tank? V=(phi)r^2h V=(phi)(7^2)(20) V=3078.760801

http://www.flickr.com/photos/54177448@N00/103329515/

Objectives:** Define and use the formula for the surface area of a cone. Define and use the formula for the volume of a cone.
 * __7.5__


 * Cone:** A 3D 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 call the vertex.
 * Base:** The circular face of the cone.
 * Lateral Surface:** The curved surface of a cylinder or cone.
 * 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 it's center point.
 * Oblique Cone:** A cone that is not a right cone.
 * Slant Height of a Cone:** The radis of the sector.

Surface Area of a Right Cone: S=L+B or S=(phi)rl+(phi)r^2
 * Blue Boxes:**

Volume of a Cone: V=(1/3)Bh or V=(1/3)(phi)r^2h

Find the surface area of a right cone with the measurements: radius 8 and slant height 14. S=(phi)(8)(14)+(phi)(8^2) S=(phi)112+(phi)64 S=552.92
 * Examples:**

Find the volume of a Cone, with the same measurements as above. V=(1/3)(phi)(8^2)(14) V= 938.29

http://www.flickr.com/photos/thomashawk/101549071/

Objectives:** Define and use the formula for the surface area of a sphere. Define and use the formula for the volume of a sphere.
 * __7.6__


 * [|Sphere]:** The set of points in space that are quidistant from a given point known as the center of the sphere.
 * Annulus:** The region between 2 circles in a plane that have the same center but different radii.

Volume of a Sphere: V=(4/3)(phi)r^3
 * Blue Boxes:**

Surface area of a Sphere: S=4(phi)r^2

A hot air balloon has a radius of 28 feet when fully blown up. How many cubic feet of hot air can it hold? V= (4/3)(phi)r^3 V= (4/3)(phi)28^3 V= (4/3)(phi)21952 V= 91952.32 cubic feet
 * Examples:**

A hot air balloon is 55 feet in diameter when blown up. The cost of fabric to cover it is $3.00 per square foot. How much is the fabric gonna cost to cover the balloon? S= 4(phi)r^2 S= 4(phi)27.5^2 S= 4(756.25)(phi) S= 9503.31 square feet. 9503.31 sq. ft. * $3.00 = 28,509.95

http://www.flickr.com/photos/gardinergirl/347805063/

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
 * 1. volume 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 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.
 * 2. surface area of triangular prism:**
 * 3. volume 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
 * 4. surface area of pyramid:**

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.
 * 5. volume 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
 * 6. surface area of cylinder:**

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.
 * 7. volume 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.
 * 8. surface are of cone:**

For my birthday I want to have the biggest bounce 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.
 * 9. volume of sphere:**

My mom is gonna get me the second ball but she has decided to wrap it. She 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
 * 10. surface area sphere:**