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Objectives
1)Explore ratios of surface area to volume. 2)Develope the concepts of maximizing volume and minimizng surface area.

**Surface Area and Volume**
The surface area, //S//, and volume, //V//, of a right rectangular prism with length with length, //l//, width, //w//, and height h are S 2//lw// + 2//wh// + 2//lh// and V//lwh//. The surface area, S, an volume, V, of a cube with side, //s,// are S 6//s//² and V//s//³

**Example**
There are two square Pizza hut boxes with the dimensions shown on the right. Which block has the greater surface area which will have more volume http://flickr.com/photos/looking4poetry/262824718/

**Solution**
Both boxes have a volume of 160 cubic inches. The surface area of box A is 2(8)(5) + 2(4)(5) + 2(4)(8) = 184 square inches. The surface area of box B is 2(10)(8) + 2(2)(8) + 2(2)(10) = 232 square inches. Box B has the greater surface area.

=__**7-2**__=

**Objectives**
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 develope a formula for the volume of a right or oblique prism. Altitude - A segment that has endpoints in the planes containing the bases and that is perpendicular to both planes. Height - A prism is the length of an altitude.

**Surface Area**
The surface area, s, of a right prism with lateral area L, base area B, perimeter p, and height h is S =L + 2B or S= hp + 2B.

**Example**
The net for a right triangular prism. What is the surface area.

**solution**
The area of each base is

B =1/2(2)(21)= 21 The perimeter of each base is p =10 + 21 + 17 48, so the lateral area is L= hp = 30(48) 1440. Thus, the surface area is S =L+ 2B 1440 + 2(21)= 1440 + 42 1482.

**Cavalieri's Principle**
If two solids have equal heights and the cross sections sections formed by every plane parallel to the bases of both solids have equal areas, the the two solids have equal volumes.

**Volume of a prism**
The volume, V, of a prism with height //h// and base area B is V = Bh. =**7-3**=

**Objectives**
· Define and use the formula for the surface area of a regular pyramid. · Define and use the formula for the volume of a pyramid. Pyramid - A polyhedron consisting of a base. Base - Is a polygon. Lateral faces - Three or polygons. Vertex of the pyramid - Triangles that share a single vortex. Base edge - Each lateral face has one edge in common with the base. Lateral edge - Intersection of two lateral faces. Regular pyramid - Pyramid whose base is a regular polygon and whose lateral face are congruent isosceles triangular. Slant height - The length of an altitude of a lateral face of a regular pyramid.

**Surface Area of a Regular Pyramid**
The surface area, S, of a regular pyramid with lateral area L, base area B, perimeter of the base p, and slant height l is S =L + B or S= 1/2lp + B.

**Example 2**
The roof of a gazebo is a regular octagonal pyramid with a base edge of 4 feet and a slant heaight of 6 feet. Find the area of the roof. If roofing material costs $3.50 per square foot, find the cost of covering the roof with this material.

**[[image:gazebo.jpg]]http://flickr.com/photos/maitri/462828373/ Solution**
The area of the roof is the lateral area of the pyramid. L =1/2lp 1/2(6)(8 * 4)= 96 square feet 96 square feet * $3.50 per square foot = $336.00

**Volume of a Pyramid**
The volume, V, of a pyramid with height h and base area B is V = 1/3Bh. =__**7-4**__=

**Objectives**
· Define and use the formula for the surface area of a right cylinder. · Define and use the formula for the volume of a cylinder. Cylinder - A solid that consists of a circular region and its translated image on a parallel plane. Lateral Surface - Connects the circles. Bases - Faces formed by the circular region and its translated images. Altitude - A segment that has endpointsin the planes containing the bases and is perpendicular to both planes. Height - Cylinder is the length of an altitude. Axis - A cylinder is the segment joining the centers of the two bases. Right cylinder - If the axis of a cylinder is perpendicular to the base. Oblique cylinder - If its not perpendicular.

**Surface Area of a Right Cylinder**
The surface area, S, of a right cylinder with lateral area L, base area B, radius r, and height h is S = L + 2B or S 2(3.14)rh + 2(3.14)r².

**Example**
A penny is a right cylinder with a diameter of 19.05 millimeters and a thickness of 1.55 millimeters. Ignoring the raised design, estimate the surface area of a penny.

**Solution**
The radius of a penny is half of the diameter, or 9.525 millimeters. Use the formula for the surface area of a right cylinder. S = 2(3.14)rh + 2(3.14)r² S = 2(3.14)(9.525)(1.55) + 2(3.14)(9.525)² 663.46 square millimeters

**Volume of a Cylinder**
The volume, V, of a cylinder with radius r, height h, and base area B is V =Bh or V= (3.14)r²h. =__**7-5**__=

**Objectives**
· Define and use the formula for the surface area of a cone. · Define and use the formula for the volume of a cone. Cone - A three-dimensional figure that consists of a circular base. Base - circular face of a cone Lateral surface - Connects the base to a single point not in the plane of the base. Vertex - Single point in the plane of the base. Altitude - Perpendicular segment from the vertex to the plane of the base. Height - Length of the altitude. Right Cone - If the altitude of a cone intersects the base of the cone at its center. Oblique Cone - If its not ^

**Example**
Find the surface area of a right cone with the measurements. http://flickr.com/photos/calvo/2068202/ The circumference of the base is c = 2(3.14)r 14(3.14). The lateral area is a sector of a circular region with circumference C = 2(3.14)l 30(3.14). The portion of the circular region occupied by the sector is c/C = 14(3.14)/30(3.14) 7/15. Calculate the area of the sector (lateral area). 3.14L² = 225(3.14) L = 7/15 · 225(3.14) 105(3.14) Calculate the base area and add the lateral area. B = (3.14)r² 49(3.14) B + L =49(3.14) + 105(3.14) 154(3.14)= 483.8
 * Solution**

**Surface Area of a Right Cone**
The surface area, S, of a right cone with lateral area L, base of area B, radius r, and slant height l is S =L + B or S= (3.14)rl + (3.14)r²

**Volume of a Cone**
The volume, V, of a cone with radius r, height h, and base area B is V =1/3Bh or V= 1/3(3.14)r²h. =**__7-6__**=

Objectives
· Define and use the formula for the surface area of a sphere. · Define and use the formula for the volume of a sphere. Sphere - The set of all points in space that are distance, r, from a given point known as the center of the sphere. Annulus - Ring shaped figure in a cylinder.

Volume of a Sphere
The volume, V, of a sphere with radius r is V = 4/3(3.14)r³

Example
The envelope of a hot-air balloon has a radius of 35 feet when fully inflated. Approximately how many cubic feet of hot hot air can it hold? http://flickr.com/photos/ray27/190352060/

Solution
V = 4/3(3.14)r³ = 4/3(3.14)(35)³ = 4/3(28,542)(3.14) = 39,245(3.14) cubic feet = 102,478 cubic feet

Surface Area of a Sphere
The surface area, S, of a sphere with radius r is S = 4(3.14)r². =**__7-7__**=

Objectives
· Define various transformations in three-dimensional space. · Solve problems by using transformations in three-dimensional space.

**Example**
You are giving AB with endpoints a - (0,5,0) and B(0,5,5). Sketch describe, and give the dimensions of the figure that results when, a) AB is rotated about the z-axis. b) AB is rotated about the y-axis. 1) Volume of triangular prism - A fish tank shaped like triangular box has dimensions of 75 x 35 x 4 feet given that 1 gallon =0.134 cubic feet, how many gallons can fit in the fish tank? 1 gallon of water= 8.33 pounds, how much will the water weigh? Solution - The volume of the fish tank is found by using the volume formula. V =Bhttp://flickr.com/photos/tgigreeny/145336883/= lwh =(75)(35)(4)= 10,500 cubic feet. The volume in gallons, divide by 0.134 V =10,500 divided by 0.134= 78,358. To approximate the weight you go weight =(78,358)(8.33)= 652722.14 pounds. 2) Surface area of triangular prism - The area of each base is B =1/2(4)(25)= 25 The perimeter of each base is p =12 + 25 + 19= 56=So the lateral area is L =hp= 42(56) = 2352 A Gas tank in a car has a length of 3 feet and an outer diameter of 1 foot and a wall thickness of about 1 inch. what is the volume of the tank at 2 gallons a car tank how many car tanks could be filled? V =(pie)r²h= pie(3.833)²(28,667) = 1323 cubic feet convert to cubic gallons.

3) Volume of pyramid - An egyption pyramid has a base edge of 850 feet and a height of 520 feet. The clay used to build was 198 pounds per cubic foot. What is the weight of the pyramid? Solution - V =1/3Bh= 1/3(850²)(520) =cubic feet. The weight in pounds is that number x 198 pounds per cubic foot= pounds or you can put it in tons, divide by 2000 4) Surface area of pyramid - The slant height is q and the base edge is r and the base area B. Solution the surface area is the sum of the lateral areas and the base area. r =q + B. r= 4(1/2rq) + r ². Can be written as R =1/2q(4r)= r².Because 4r is the perimeter of the base. R = 1/2qp + r ². 5) volume of cylinder - A Gas tank in a car has a length of 3 feet and an outer diameter of 1 foot and a wall thickness of about 1 inch. what is the volume of the tank at 2 gallons a car tank how many car tanks could be filled? V =(pie)r²h= pie(3.833)²(28,667) = 1323 cubic feet convert to cubic gallons. 6) Surface area of cylinder - A pop can with a diameter of 112 millimeters and a thickness of 112 millimeters. Solution - The radius of a pop can is half of the diameter or 56 millimeters, the formula is S =2(3.14)rh + 2(3.14)r². S= 2(3.14)(56)(112) =2(3.14)(56)²= 663.46 square millimeters. 7) Volume of cone - The cone shaped hill in my back yard has a radius of s 1 mile and a height of u s 0.5 miles to find the volume you do V =1/3(pie)s²u= 1/3(1²)(0.5) = cubic miles 8) Surface area of cone - There is a cone with a height L of 21 and a diameter of 10. The circumference of the base is C =2(pie)r= 14 (pie). Lateral area of the circular region with circumference C =2(pie)L= 30(pie). The portion of the circular region occupied by the sector is c/C =14(pie)/30(pie)= 10/21. Calculate (pie)L² =225(pie) L= 10/21 x 225(pie) =138(pie). Calculate the base area and add the lateral area B= (pie)r² =62(pie). B= L =62(pie) + 162(3.14)= 224(pie) = 703.36 9) Volume of a sphere - The ball on the top of a New York building has a radius of 15 feet how many cubic feet of air can it hold? solution - 4/3(pie)³ =4/3(3.14)(15)³= 4/3(3375)(pie) =10597.5 (pie) cubic feet= 33276 cubic feet 10) Surface area of a sphere - The ball has a diameter of 30 feet so the radius is 15. S =4(pie)r²= 4(pie)(15)² =4(225)(pie)= 706.5(pie) = 2218.41 square feet.