loeder

7.1 Surface area and volume Rormulas** The surface area, S, and the Volume, V, pf a right Rectangular prism with lengh l, and height, h, are S=2lw+2wh+2lh and V=lwh The Surface area, S, and Volume, V, of a cube with sides are S=6s² and V=s³
 * lode35 Ch. 9 link

Altitude-** is the perpendicular segment from the vertex of the plane of the base
 * 7.2
 * Height**- length of the altitude

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

Solution:** The area of each base is: B=1/2(2)(21)=21 The perimeter for each base is: P=10+21+17=48 so the lateral area is: L=HP=30(48)=1440 Thus, the SA is: S=L+2B=1440+2(21)=1440+42=1482
 * Example 1- The net for a right triangular prism is below. What is the Surface area?

V=BH=LWH=(110)(50)(7)=38,500 cubic feet To approximate the volume in gallons, divide by 0.134. V=38,500÷o.134 is about 287,313 gallons To approximate the weight, multiply by 8.33. Weight is approximately (287,313)x(8.33) wich is approximately 2,393,317 pounds.
 * Example 2- An aquarium in the shape of a right rectangular prism has the dimensions of 110x50x7 feet. Given that one gallon is about 0.134 cubic feet, how many gallons of water will the aquarium hold? Given that one gallon of water is about 8.33 pounds, how much will the water weigh?**
 * Solution:** The volume of the aquarium is found by using the volume formula.

B=1/2 AP=1/2(84)(7 radical 3)=294 radical 3 is approximately509.22 square inches The volume is: V=BH=(294 radical 3)(48)=14112 radical 3 is approximately 24,443 cubic inches.
 * Example 3- An aquarium has the shape of a right regular hexagonal prism with the dimensions shown at right. find the volume of the aquarium.**
 * Solution:** The base of the aquarium has a perimeter of (14)(6), or 84, inches and an apothem of 7 radical 3 inches, so the base area is found as follows:
 * Cavalieri's Principle-** If two solids have equal heights and the cross sections formed by every plane parallel to the bases of both solids have equal areas, then the two solids have equal volumes.
 * 7.3**
 * 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 bottom of the figure.
 * Lateral Faces-** The faces of a prism or pyramid that are not bases.
 * Vertex Of The Pyramid-** A point where the edges of the pyramid intersect plural.
 * Base Edge-** An edge that is at the base of the pyramid.
 * Lateral Edge-** The intersection or two lateral faces of a polyhedron
 * Altitude-** is the perpendicular segment from the vertex of the plane of the base
 * Height**- length of the altitude
 * Regular Pyramid-** A pyramid whose base is a regular polygon and whose lateral faces are congruent isosceles triangles.
 * Slant Height-** In a regular pyramid the lengh of an altitude of a lateral face.

Solution:** The Surface area is the sum of the lateral ares and the base area. S=L+B S=4(1/2 SL)+sPi ² This can be rewriten as follows: S=1/2L(4s)+sPi ² Because 4 is the perimeter of the base, S=1/2LP+sPi ²
 * Example 1- Find the surface area of a regular square pyramid whose slant height is L, and whose base edge length is s.

S=L+B S=4(1/2SL)+S² or S=1/2L(4S)+S² S=1/2Lp+S²
 * To Find the surface area of a regular squre pyramid whose slant height is L and whose base edge length is S.**

The Surface area, S, of a regular pyramid with the Lateral area, L, Base area,B, perimeter of the base, p, and slant height L is S=L+B or S=1/2Lp+B
 * Surface area of a regular pyramid**

The Volume, V, of a pyramid with height, H, and Base, B, is V=1/3BH
 * Volume of a Pyramid**
 * 7.4 Surface area and Volume of cylinders**
 * Cylinder-** A solid that consists of a region and its translated image in a parallel plane with a lateral surface connecting the circles.
 * Lateral Surface-** The curved surface of a cylinder or cone.
 * Bases-** The top and bottom of the figure.
 * Altitude-** is the perpendicular segment from the vertex of the plane of the base.
 * Height**- length of the altitude.
 * Axis-** A cylinder that the segment joining the center of the two bases.
 * Right Cylinder-** A cylinder whose axis is perpendicular to the bases.
 * Oblique Cylinder-** A cylinder that is not a right cylinder.

The Surface area, S, of a rigt cylinder with lateral area, L, Base area, B, Raduis, R, Height, H, is S=L+2B or S=2Pi rh+2Pi r
 * Surface Area of a Right Cylinder**

The volume, V, of a cylinder with radius, R, Height, H, and Base are, B, is V=Bh or V=Pi ² Cone-** A cone is a 3-D figure that consists of a circular baseand a curved lateral face that connects the base to a single point not in the plane of the base, called the vertex. Oblique Cone- A cone that is not a right cone.
 * Volume of a Cylinder**
 * 7.5 Surface area and Volume of Cones
 * Base-** The base is a circle.
 * Lateral Surface-** The lateral face connects the base to the vertex.
 * Altitude-** The altitude of the cone is the perpendicular segment from the vertex to the plane of the base.
 * Height-** The height of the cone is the length of the altitude.
 * Right Cone-** If the altitude of the cone intersects the base of the cone at its center, the cone is a right cone.
 * Slant height of a Cone-** When you have a net for a cone, the lateral face creates the area of Pi, This is called the sector. The radius of the sector is the slant height.

The Surface area, S, or a right cone with lateral area, L, base of area, B, radius, R, and slant height, L, is S=L+B or S=Pi rl=Pi r²
 * Surface area of a right cone**

The Volume, V, of a cone with radius, R, height, H, and base area, B, is V=1/3bh or V=1/3Pi r²h
 * Volume of a cone**
 * 7.6 Surface area and volume of spheres**
 * Sphere-** The set of points in space that are equidistant from given.
 * Annulus-** The region between 2 circles in a plane that have the same center but diffrent radii.

The volume, V, of a sphere with radius, R, is V=4/3 Pi r³ V=4/3 Pi r³ =4/3 Pi (27)³ =4/3 (19683) Pi =26,244 Pi cubic feet approximately 82,488 cubic feet.
 * Volume of a sphere**
 * Example 1**The envelope of a hot-air ballon has a radius of 27 feet when fully inflated approximately hom many cubic feet of hot air can it hold?
 * Solution:**

The surface area, S, of a sphere with radius, R, is S=4 Pi r²
 * Surface area of a sphere**

First estimate the surface area of the inflated balloon envelope. The balloon is approximately a sphere with a diameter of 54 feet, so the radius is 27 feet. S=4 Pi r² =4 Pi(27)² =4(729) Pi =2916 Pi approximately 9160.9 square feet. Now multiply the surface area of the fabric by the cost per square foot to find the approximate cost of the fabric. 9160.9 square feet× $1.31 per square foot approxametely $12,000.
 * Example 2** The envelope of a hot-air balloon is 54 feet in diameter when inflated. The cost of the fabric used to make the envelope is $1.31 per square foot. Estimate the total cost of the fabric for the balloon envelope.
 * Solution:**