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embr222 Chapter 9 link

__Chapter 7.1__ [|Lesson 7.2] The objectives of chapter 7.1: - To explore ratios of surface area to volume - Develop the concepts of maximizing volume and minimizing surface area
 * Chapter 7.1**

__Surface Area and volume Formulas:__ //Rectangular Prism using length (l), width (w) and height (h) to find surface area and volume.// - Surface Area = S or SA ^S=2lw+2wh+2lh - Volume = V ^V=lwh

//Finding surface area and volume of a cube using s for side.// - Surface Area = S or SA ^=6s^2 - Volume = V ^V=s^3

Examples: __Maximizing Volume:__ __Ratio of Surface Area to Volume:__ SQUARES: CUBES:
 * Side of Square || Length, l || Width, w || Height, h || Volume, v ||
 * 1 || 9 || 6.5 || 1 || 58.5 ||
 * 2 || 7 || 4.5 || 2 || 63 ||
 * 3 || 5 || 2.5 || 3 || 37.5 ||
 * x || 11-2x || 8.5-2x || x || lwh ||
 * x || 11-2x || 8.5-2x || x || lwh ||
 * Length, l || Surface Area, S or SA || Volume, v || Ratio || Reduced ||
 * 1 || 6u^2 || 1u^3 || __6__ 1 = || 6 ||
 * 2 || 10u^2 || 2u^3 || __10__ 2 = || 5 ||
 * 3 || 14u^2 || 3u^3 || __14__ 3 = || 4.66 ||
 * 4 || 18u^2 || 4u^3 || __18__ 4 = || 4.5 ||
 * n || 4n+2 || n || __4n+2__ n = = || __SA__ V = =reduced ||
 * n || 4n+2 || n || __4n+2__ n = = || __SA__ V = =reduced ||
 * Side, s || Surface Area, S or SA || Volume, v || Ratio || Reduced ||
 * 1 || 6u^2 || 1u^3 || __6__ 1 = || 6 ||
 * 2 || 24u^2 || 8u^3 || __24__ 8 = || 3 ||
 * 3 || 54u^2 || 27u^3 || __54__ 27 = || 2 ||
 * 4 || 96u^2 || 64u^3 || __96__ 64 = || 1.5 ||
 * n || 6n^2 || n^3 || __6n^2__ n^3 = || __6__ n ||
 * n || 6n^2 || n^3 || __6n^2__ n^3 = || __6__ n ||

 __//The objectives of chapter 7.2://__ - Define and use a formula for finding the surface area (s or sa), and volume (v), of a right prism. - Use Cavalieri's Principle to develop a formula for the volume of a right or oblique prism.
 * Chapter 7.2**

[|Right Triangular Prism]

//__Words To Know:__// - Altitude: Its a segment that has endpoints in the planes containing the bases and that is perpendicular to both planes. - Height: Is the length of the altitude

//__Surface Area and Volume Formulas:__// - Surface Area - S or SA ^S= L+2B or S=hp+2B - Volume = V ^V=BH

//__Cavalieri's Principle:__ -// If two solids have equal heights and the cross sections formed by every plane parellel to the bases of both solids have equal areas, then the two solids have equal volumes.

//__The objectives of chapter 7.3:__// - Define and use a formula for the surface area of a regular pyramid. - Define and use a formula for the volume of a pyramid.
 * Chapter 7.3**

//__Words To Know:__// - 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 polygonal face that is opposite the vertex. - 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 - 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 teh vertex perpendicular to the plane of the base. - Height - The length of an altitude of a polygon. - Regular pyramid - A pyramid whose base it a regular polygon and whose lateral faces are congruent isosceles triangles. - Slant height - In a regular pyramid, the length of an altitude of a lateral face.

//__Surface Area and Volume Formulas:__// b =base; p= perimeter; L = [|Slant Height] - Surface Area - S or SA ^ S = 1/2lp+b - Volume - V ^ V = 1/3Bh

Regular Pyramid [|Indepth picture of a regular pyramid]





//__Examples:__// - The base edge is 4 and the slant height is 6. Find the surface area of the pyramid. ^ L =1/2 lp= 1/2(6)(8x4) = 96 square units - The base edge is 776 and the height is 481. Find the volume of the pyramid. ^ V = 1/3 Bh =1/3(776^2)(481)= 96,548,885 cubic units. //__Objectives in chapter 7.4:__// - Define and use a formula for the surface area of a right cylinder. - Define and use a formula for the volume of a cylinder
 * Chapter 7.4**

//__Words To Know:__// - Cylinder: A solid that consists of a circular region and its translated image in a parallel plane with a lateral surface connecting the circl - Lateral Surface: The curved surface of a cylinder or cone. - Bases: The faces formed by the circular region and its translated image. - Altitude: A segment that has end points in the planes containing the bases and is perpendicular to both planes. - Height: The length of an altitude of a polygon. - Axis: The segment joining the cinters of the two bases - Right Cylinder: A cylinder whose axis is perpendicular to the bases. - Oblique Cylinder: A cylinder that isn't a right cylinder.

//__Formulas for Surface Area and Volume of a Right Cylinder:__// - Surface Area: ^ S = 2rh+2r2 - Volume: ^ V = r^2h

[|Right Cylinder]

//__Examples:__// - Find the surface area of a cylinder with a diameter of 19.05, a hieght of 1.55. Find the surface area of the cylinder. S = 2rh+2r^2 S =2(9.526)(1.55) + 2(9.525)^2= 663.46 square units.

- Find the volume of a cylinder with a diameter of 2, a height of 6. Find the volume of the cylinder. V = r^2h V =1^2(h)= 1^2(6) = 6 V = 18.84 cubic units

//__Objectives in Chapter 7.5:__// - Define and use the formula for the surface area of a cone. - Define and use the formula for the volume of a cone.
 * Chapter 7.5**

//__Words To Know:__// - Cone: A 3-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. - Base: The circular face of the cone. - Lateral Surface: It connects the base to a single point not in the plance of the base, called the vertex. - 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 and its center point. - Oblique Cone: A cone that isn't a right cone. - Slant Height of a Cone: Hieght of a cone. - Circumference: The distance around the circle.

[|Cone]

//__Formulas for Surface Area and Volume of a Right Cone:__// - Surface Area: ^ S =L+B or S= rl+r^2 - Volume: ^ 1/3 bh - Circumference = C

//__Examples:__// - Find the surface area with the given information. ^ C of base =2r= 14 ^ lateral area with c =2nl= 30 ^ The proportion of the circular region occupied by the sector is __c__ = __7__ C 15 ^ Find the area of the sector (lateral face) l^2 = 225 L =__7__ (225)= 105 15 ^ Find the base area and add the lateral area. B =r^2= 49 B + L =49 + 105= 154 = 483.8

- The radius is 5, the hieght is 2. Find the volume. ^ V =1/3r^2h= 1/3 (1^2)(0.5) = 52.4 cubic units

//__Objectives in Chpater 7.6:__// - Define and use the formula for the surface area of a sphere. - Define and use the formula for the volume of a sphere.
 * Chapter 7.6**

//__Words To Know:__// - Sphere: A set of points in space that are the same distance from a given point known as the center fo the sphere. - Annulus: The regio between two circles in a plane that have the same center but different radii.

//__Formulas for Surface Area and Volume:__// - Surface Area: ^ S = 4r^2 - Volume: ^ V = 4/3r^3

[|Real Sphere] [|Indepth Sphere]

//__Examples:__// - The diameter is 54 and the radius is 27. Find the surface area. ^ S = 4r^2 =4(27)^2= 4(729) =2916= 9160.9 square units. - The radius is 27. Find the volume. ^ V = 4/3^3 =4/3(27)^3= 4/3(19,683) =26,244= 82,488 cubic units.