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•Develop the concepts of maximizing volume and minimizing surface area
 * __Surface Area and Volume Formulas:__**

A surface area is the total area of all of the exposed areas of the object. Volume of a object is the numbure of cubes that are in the figure.

Th//e// volume formula is: S=2lw+2lh and V=lwh

Side of Square: Length Width Height Volume 1in. 9 6.5 1 58.5 2in. 7 4.5 2 63 3 in. 5 2.5 3 37.5


 * __Surface Area and Volumme of Prisms:__**

•Define and use a forfula for finding the surface area of a right prism •Define and use a vormula 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

Altitude: An altitude of a prism is a segment that has endpoints in the plnes containing the ases and that is perpendicular to both planes.

Height: The height of a prism is the length of an altitude.

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

Ex 1: The net for a right triangular prism is below. What is its 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)=140 Thus, the surface area is S=L+2B=1440+2(21)=1440+2=1482

Ex 2: An aquarium in the shape of a right rectangular prism has dimentions of 110 x 50 x 7 feet. Given that 1 gallon = 0.134 cubic feet, how many gallons of water will the aquarium hold? given 1 gallon of water = 8.33 pounds, how much will the water weight?

Solution: The volume of the aquarium is found by using the volume formula V=Bh = lwh = (110)(50)(7) = 38,500 cubic feet To approximate the volume in gallons, divide by 0.134. V = 38,5000/0.134 = 287,313 gallons To approximate the weight, multiply by 8.33. Weight = (287,313)(8.33) = 2,393,317 pounds Ex 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√ 3 inches, so the base area is found as follows: B = 1/2ap = 1/2(84)(7√ 3) = 509.22 square inches, the volume is V = Bh = (294√ 3)(48) = 14112√ 3 = 24,443 cubic inches

__Cavalieri's Principle:__ If twosolids have an equal heights and the vertical section formed by every plane parallel to the bases of both of the solids have equal areas, then the two solids have equal volumes.

__Volume of a Prism:__ Formula, V = B

•Define and use a formula for the surface area of a regular pyramid •Define and use a formula for the volume of a pyramid
 * __Surface Area and Volume of Pyramids:__**

__A Pyramid:__ A polyhedron in which all but one of the polygonal faces intersect at a single point known as the vertex of the pyramid

__A Base:__ An edge that is part of the base of a pyramid, each lateral face has one edge in common with the base

__A Lateral Face:__ The faces of a prism or pyramid that are not bases

__A Vertex of the Pyramid:__ A point where the edges of a figure intersecs plural, vertices

__A Base Edge:__ Anyside of a parallelogram

__A Lateral Edge:__ The intersection of two lateral faces of a polyhedron

__A Altitude:__ A segment from the vertex perpendicular to the plane of the base

__A Height:__ The length of an altitude of a polygon

__A Regular Pyramid:__ A pyramid whose base is a regular polygon and whose lateral faces are congruent isosceles vriangles

__A Slant Height:__ In a regular pyramid, the length of an altitude of a lateral face

Ex 1: Find the surface area of a regular square pyramid whose slant height is L and whose base edge length is S.

Solution: The surface area is the sum of the lateral areas and the base area. S = L + B S = 4 (1/2sl) + s² This can be rewritten as follows: S = 1/2lp + s²

Surface Area of a Regular Pyramid: Formula, 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 height of 6 feet. Find the area of the foot, find the cost of covering the roof with this material.

Solution: The area of the roof is the lateral area of the pyramid. L = 1/2lp = 1/2(6)(8x4) = 96 square feet, 96 square feet x $3.50 per square foot = $336.00

__Volume of a Pyramid:__ Formula, V = 1/3Bh

Example 3: The pyramid of Khufu is a regular square pyramid with a base edge of approximetely 776 feet and an original height of 481 feet. The limestone used to construct the pyramid weighs approximately 167 pounds per cubic foot. Estimate the weight of the pyramid of Khufu. (Assume the pyramid is solid)

Solution: The volume of the pyramid is found as follows: V = 1/3Bh = 1/3(7762)(481) 96,548,885 cubic feet The weight is pound is 96,548,885 cubic feet x 167 pounds per cubic foot = 16,123,663,850 pounds, or 8,061,831 tons.


 * __Surface Area and Volume of Cylinders:__**

•Define and use a formula for the surface area of a right cylinder •Define and use a formula for the volume of a cylinder



__A Cylinder:__ a solid that consists of a circular region and its translated image in a parellel plane with a lateral surface connecting the circles

__A Lateral surface:__ the curred surface of a cylider or cone

__A Bases:__ the faces formed by the circular region and its translative image

__A Altitude:__ A segment that has endpoints in the planes containing the bases and is perpendicular to both planes

__A Height:__ the lenght of an altitude of a polygon

__A Axis:__ The segment joining the centers of two bases

__A Right cylinder:__ A cylinder whose axis is perpendicular to the bases

__A Oblique cylinder:__ A cylinder that is not a right cylinder

__Surface Area of a Right Cylinder:__ Formula, S = L + 2B or S = 2πrh + 2πr²

Ex 1: A penny is a right cylinder with a diameter of 19.05 millimeters and a thickness of 1.55 millimeters. Ignoring the raised design, extimate the surface 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 = 2πrh + 2πr² S = 2π(9.525)(1.55) + 2π(9.525)² = 663.46 square millimeters

__Volume of a Cylinder:__ Formula, V = Bh or V = πr²h

Ex 2: The tank has a length of 31 feet 6 1/2 inches and an outer diameter of 8 feet 0 inches. Assuming a wall thinkness of about 2 inches, what is the volume of the tank? At 15 gallons per car, how many car tanks could be filled from the storage tank if it starts out completely full of gasoline?

Solution: The tank is not perfectly cylindrial, because of its hemispherical heads, but you can approximate its volume by a slightly shorter cylinderical tank, 29 feet long. Subtracting the wall thickness from the dimensions of the tank, V = π r²h = π1323 cubic feet 1323 cubic feet x 7.48 gallons per cubic foot = 9896 gallons 9896/15 or = 660, 15-gallon fill-ups


 * __Surface Area and Volume of Cones:__**

•Define and use the formula for the surface area of a cone •Define and use the formula for the volume of a cone



__A Cone:__ A three 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

__A Base:__ The circular face of the cone

__A Lateral Surface:__ The curved surface of a cylinder or cone

__A Altitude:__ A segment from the vertex perpendicular to the plane of the base

__A Height:__ The length of an altitude of a polygon

__A Right cone:__ A cone in which the altitude intersects the base at its center point

__A Oblique Cone:__ A cone that is not a right cone

__A Slant height of a Cone:__ The radius, L of the sector is the slant height of the cone

A drawing of a net for a cone and lable it's radius and slant height:

Ex 1: Find the surface area of a right cone with the indicated measurments

Solution: The circumference of the base is c = π L = 14 π The lateral area is a sector of a circular region with circumference C = 2π = 30π The portion of the circular region occupied by the sector is c/C = 14π/30π = 7/15 Calculate the area of the sector (lateral area): πl² = 225π L = 7/15 x 225π = 105π Calculate the base area and add the lateral area: B = πr² = 49π B + L = 49π + 105π = 154π = 483.8

Surface Area of a right cone: Formula, S = L + B or S = πrl + πr²

Volume of a Cone: Formula, V = 1/3Bh or V = 1/3πr²h

Ex 2: A volcanologist is studying a violent eruption of a cone-shaped valcano. The original vlcanic cone had a radius of 5 miles and a height of 2 miles. The eruption removed a cone-shaped area from the top of the colcano. this cone had a radius of 1 mile and a geight of 1/2 mile. What percent of the total volume of the original volcano was removed by the eruption?

Soluution: Find the volume of the origimal volcano: V = 1/3 πr2h = 1/3(5²)(2) = 52.4 cubic miles Find the volume of the desroyed cone: V = 1/3πr2h = 1/3(1²)(0.5) = 0.52 cubic miles Find the percent of the original volcano removed by the eruption: (0.52/52.4)100 = 1%

•Define and use the formula for the surface area of a sphere •Define and use the formula for the volume of a sphere
 * __Surface Area and Volume of Spheres:__**



A Sphere is the set of all points in space that are the same distance, from a given point known as the center of the spere

A Annulus is the region between two circles in a plane that have the same center but different radii

The Formula of a Volume of a Sphere is: V = 4/3πr³

Ex 1: The envelope of a hot-air balloon has a radius of 27 feet when fully inflated. Approximately how many cubic feet of hot air can it hold?

Solution: V = 4/3πr³ V = 4/3π(27)³ V = 4/3(19,683)π V = 26,244π cubic feet = 82,488 cubic feet

Surface area of a sphere: Formula, S = 4πr²

Ex 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 evelope.

Solution: First estimate the surface area of the inflated balloon envelope. The balloon is approzimately a sphere with a diameter of 54 feet, so the radius is 27 feet. S = 4πr² S = 4π(27)² S = 4(729)π S = 2916π = 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 x $1.31 per square foot = $12,000