3rammar

__//Objectives//__:
 * __[[image:school41.gif]]7.__1 Surface Area and Volume**
 * //__Explore ratios of surface area to volume__//
 * //__Develop the concepts of maximizing volume and minimizing surface area__//

The surface area of an object is the total area of all the exposed surfaces of the object. The volume of a solid object is the number of nonoverlapping unit cubes that will exactly fill the interior of the figure.

__Surface Area and Volume__

The surface area, //S//, and volume, //V//, of a right rectangular prism with lengh //l//, width //w//, and height //h// are

S = 2//lw// + 2//wh// + 2//lh// and V = //lwh//

The surface area, //S//, and volume, //V//, of a cube with side s are

S= 6//s// and V=//s//

EXAMPLE: Maximizing Volume

Example: A cereal company is choosing between tho box desings. Which design has the greater surface area and thus requires more material for the same volume?
 * side of square, x || length, l || width, w || height, h || Volume, lwh ||
 * 1 || 9 || 6.5 || 1 || 58.5 ||
 * 2 || 7 || 4.5 || 2 || 63 ||
 * 3 || 5 || 2.5 || 3 || 37.5 ||
 * x || ? || ? || ? || ? ||

Boxes A & B

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 Surface Area and Volume of Prisms**

__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.__

An **"//altitude"//** of a prism is a segment that has endpoints in the planes containing thebases and that is perpendicular to both planes. The //**"height"**// of a orims is the length of an altitude.

__NOTES:__ The surface area of a prism may be broken down into two parts: 1) The area of the bases 2) The are of the lateral faces.


 * __Surface Area of a Right Prism__

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 1:

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//.

__Volumes of Oblique Prism__

In an oblique prism, the lateral edges are not perpendicular to the bases, and there is no simple general formula fot the surface area.




 * __Cavalieri's Principle__

If tho solids have equal heights and the cross section formed by every plane parallel to the bases of the both solids have equal areas, then 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 Surface Area and Volume of Pyramids**

__Objectives:__
 * __Define and use a formula for the surface area of a rectangular pyramid.__
 * __Define and use a formula for the volume of a pyramid.__

__NOTES:__

A "pyramid" consits on: 1)base 2)lateral faces 3)vertex of the pyramid 4)base adge 5)lateral edge

A __**"regular pyramid"**__ is a pyramid whose base is a regular polygon and whose lateral faces are congruent isoceles triangles.

__Example 1:__

The surface area is the sum of the lateral areas and the base area.

S= L + B S= 4(1/2sl) + s

Because 4s is the perimeter of the base,

S= 1/2lp + s


 * __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= 1/2 lp + B// ||


 * __Volume of a Pyramid__

The volume, //V,// of a pyramid with height h and base area //B// is:

//V= 1/3 Bh// ||

__Example 2:__ The volume of the pyramid is found as follows: V= 1/3 Bh = 1/3 (776)(481) = 96,548,885 cubic feet.


 * 7.4 Surface Area and Volume of Cylinders**

__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.__

plane, with a lateral surface connecting the circles.
 * Cylinder;** is a solid that consist of a circular region and its translated image on parallel


 * Bases;** the faces formed by the circular region.
 * Altitude**; is a segment that has endpoints in the planes.
 * Axis;** is the segment joining the center of the two bases.


 * __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~rh + 2~r// ||


 * __Volume of a Cylinder__

The volume, V, of a cylinder with radius r, height h, and base area B is;

//V= Bh// //or V= ~rh// ||




 * 7.5 Surface Area and Volume of Cones**

__Objectives__:
 * __Define and use the formula for the surface area of a cone.__
 * __Define and use the formula for the volume of a cone.__

__NOTES:__

A **"cone"** is a three- dimensional figure that consist of a circular **"base"** and curved
 * "lateral face"** that connects the base to a single point in the plane of the base, called de
 * "vertex".**


 * Altitude;** perpendicular segment form the vertex to the planeof the base.
 * Height;** the leght of the altitude.

__Examples:__

Math - Geometry
 * Geometric Formulas**

Right Cone



Solving for lateral surface area:



Solution
 * radius (r) ||  || units ||
 * height (h) ||  || units ||
 * lateral surface area (alateral) || = || 20.116008064342 || units2 ||


 * __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= ~rl + ~r// ||


 * __Volume of a Cone__

The volume, //V//, of a cone with radius //r,// height //h,// and base arrea //B// is;

//V= 1/3 Bh// ||




 * 7.6 Surface Area and Volume of Spheres**

__Objectives:__
 * __Define and use the formula for the surface area of a sphere.__
 * __Define and use the formula for the volume of a sphere.__

(The height and diameter of the cylinder are the same as the sphere's diameter.)
 * Sphere**: is the set of all points in space that are the same distance, r, from a given point known as the center of the sphere.



The [|surface area] of a sphere of radius //r// is 

Its enclosed [|volume] is 

//__Example 1:Volume of a Sphere__//

The envelope of a hot-air ballon has a radius of 27 feet when fully inflated. Approximately how many cubic feet of hot air can it hold?

//**SOLUTION;**//  = 4/3 ~(27)3 = 4/3(19,683)~ =26,244~ cubic feet= 82,488 cubic feet //__Example 2: Area of a Sphere__//

First estimate the surface area of the inflated ballon envelope. The ballon is approximately a sphere with diameter of 54 feet, so the radius is 27 feet.

//**SOLUTION;**//  = 4~(27)2 = 4(729)~ =2916~ 9160.0 square feet now multiply the surface area of the fabric by the cost per square foot to find the approximate cost of the fabric. 9160.0 square feet x $1.31 per square foot ~ $12,00O=

= 7.7 Three- Dimensional Symmetry.= = = =Objectives:= = =
 * __Define various transformations in three-dimesional space.__
 * __Solve problems by using transformations in three-dimensional space__.


 * NOTES:**

//(A three-dimensional figure may be reflected across a plane, just as two-dimensional figure can be reflected acroos a line.)//




 * REVOLUTIONS IN COORDINATE SPACE**.

To define **cylindrical coordinates**, we take an axis (usually called the **//z//-axis**) and a perpendicular plane, on which we choose a ray (the **initial ray**) originating at the intersection of the plane and the axis (the **origin**). The coordinates of a point //P// are: the polar coordinates (//r//,) of the projection of //P// on the plane, and the coordinate //z// of the projection of //P// on the axis




 * __SOMETHING NEW!__**

Any combination of reflections along the X, Y and Z axes may be used. For example, 101 for (I2) will cause reflection along axes X and Z, and 111 will cause reflection along axes X, Y and Z. When combinations of reflections are requested, the reflections are done in reverse alphabetical order. That is, if a structure is generated in a single octant of space and a GX card is then read with I2 equal to 111, the structure is first reflected along the Z-axis; the structure and its image are then reflected along the Y-axis; and, finally, these four structures are reflected along the X-axis to fill all octants. This order determines the position of a segment in the sequence and, hence, the absolute segment numbers.