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7.1 Surface Area and Volume Objective-** Explore ratios of Surface Area to Volume. Rectangular Prism:** S= 2LW+2WH+2LH and V= LWH Solutions:** Both boxes have a volume of 160 cubic inches. The SA of both box **A** is 2(8)(5)+2(4)(5)+2(4)(8)=184 The surface area of box **B** is 2(10)(8)+2(4)(8)+2(2)(10)=232 square inches. Box **B** has a greater SA.
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 * Objective-** Develop the concepts of maximizing Volume and minimizing Surface Area.
 * //Surface Area (SA) and Volume (V) formulas-//
 * Cube:** S=6s^2 and V=s^3
 * Example 1- A cereal companyis choosing between two box designs with the dimensions A.)5x4x8 B.)8x2x10. Which design has the greater SA and thus requires more material for the same V?

Objective-Define and use a formula for finding the SA and V of a Right Prism. S=L+2B or S=HP+2B 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__ 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. 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.
 * 7.2 Surface Area and Volume of Prisms**
 * Objective-**Use Cavalieri's Principle to develop a formula for the V of a Right or Oblique Prism.
 * Altitude-** An altitude of a prism is a segment that has end points in the places containing the basses and that is perpendicular to both planes ( or the absolute height of a prism. )
 * Height-** The height of a prism is the length of the altitude.
 * SA of right** **prism-** The SA of a right prism, S, with lateral area L, base area B, perimeter P, and Height H is
 * Example 1- The net for a right triangular prism is below. What is the SA?**
 * Solution:** The area of each base is: B=1/2(2)(21)=21
 * 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.
 * 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//.
 * Volume of a Prism-** The volume, V, of a prism with height H and basearea B is: V=BH

Objectives-** Define and use a formula for the SA and V of a regular pyramid. Solution:** The SA is the sum of the lateral ares and the base area. S=L+B S=4(1/2 sL)+s^2 This can be rewriten as follows: S=1/2L(4s)+s^2 Because 4s is the perimeter of the base, S=1/2LP+s^2 L=1/2//L//P=1/2(6)(8x4)=96 Square feet 96 square feet x $3.50 per suare foot=$336.00 V=1/3BH which is approximately 1/3(776^2)(481) which is approximately 96,548,885 cubic feet. The weight in pounds is 96,584,885 cubic feet x 167 pounds per square foot, which is approximately 16,123,663,850 pounds, or 8,061,831 tons.
 * 7.3 Surface Area and Volume of Pyramids
 * Pyramid-** A pyramid is a polyhedron that consists of a base, which is a polygon, and three or more lateral faces.
 * Base-** The base is a polygon.
 * Lateral Face-** The lateral faces are triangles that share a single vertex.
 * Vertex of the Pyramid-** the vertex of a pyramid is where all of the lateralfaces meet at one point.
 * Base Edge-** The edge in common with the lateral face and the base.
 * Altitude-** The altitude of a pyramid is the perpendicular segment from the vertex to the plane of the base.
 * Height-** The height of a pyramid is the length of altitude.
 * Regular Pyramid-** A regular pyramid is a pyramid whose base is a regular polygon, and whose lateral faces are congruent isosceles triangles.
 * Slant Height-** The slant height is the length of an altitude of a lateral face of a regular pyramid.
 * Example 1- Find the surface area of a regular square pyramid whose slant height is L, and whose base edge length is s.
 * SA of a regular pyramid-** The Surface Area, S, of a regular pyramid with lateral area L, base B, perimeter of the base P, and the slant height //L// is: S=L+B or S=1/2//L//P+B
 * Example 2- The roof of a gazebo is a octogonal pyramid witha base edge of 4 feet and a slant height 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 woth this material.**
 * Solution:** The area of the roof is the lateral area of the pyramid.
 * Volume of a pyramid-** The volume, V, of a pyramid with height H and Base area B is: V=1/3BH
 * Example 3- The pyramid of Khufu is a regular suare pyramid with a base edge of approximately 776 feet and an original height of 481 feet. The limestone used to construct the pyramid weighs approximately 167 pouns per cubic foot. Estimate the weight of the pyramid of Khufu. ( Asume the pyramid is solid )**
 * Solution:** The volume of the pyramid is found as follows:

Objectives-** Define and use a formula for the SA and V of a cylinder. S=2(3.14)rH+2(3.14)r^2 S=2(3.14)(9.525)(1.55)+2(3.14)(9.525)^2, which is approximately 663.46 square mm.
 * 7.4 Suface Area and Volume of Cylinders
 * Cylinder-** A cylinder is a solid that consists of a cicular region and its translated image on a parallel plane, with a lateral face connecting the circles.
 * Lateral Surface-** The lateral surface area connects the base circles.
 * Bases-** The bases of the cylinder are the faces formed by the circular region and its translated image.
 * Altitude-** An altitude of a cylinder is a segment that has endpoints in the planes containing the bases and is perpendicular to both planes.
 * Height-** The height of the cylinder is the length of the altitude.
 * Axis-** The axis of a cylinder is the segment joining the centers of the two bases.
 * Right Cylinder-** If the axis of the cylinder is perpendicular to the bases, then the cylinder is a right cylinder.
 * Oblique Cylinder- If Not,** Then it is an oblique cylinder.
 * SA 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)r^2
 * Example 1- Apenny is a right cylinder with a diameter of 19.05 mm and a thickness of 1.55 mm. Ignoring the raised design, estimate the SA of the penny.**
 * Solution:** The radius of the penny is half the diameter, or 19.525 mm. Use the formula for the surface area of a right cylinder.

V=(3.14)2^2H, which is approximately (3.14)(3.833)^2(28.667) which is approximately 1323 cubic feet Convert from cubic feet to gallons. 1323 Cubic feet x 7.28gallons per cubic foot, which is approximately 9896gallons Thus the tank could deliver about 9896/15, or approximately 660, 155 gallon fill-ups.
 * Volume of a Cylinder-** The Volume of a cylinder, V, with the radius r, height H, and base area B is: V=BH or V=(3.14)r^2H
 * Example 2- A tank has a length of 31 feet 6.5 inches and an outer diameter of 8 feet. Assuming a wall thickness of 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 cylindrical, because of its hemispherical heads,But you can approximate its volume by a slightly shorter cylindrical tank, say, 29 feet long. Subracting the wall thickness from the dimensions of the tank,



Objectives-** Define and use a formula for the SA and V of a cone. Example 1-** Find the SA of the right cone with the indicated measurements. The lateral area is the sector of the circular regionwith circumference C= V=1/3(3.14)r^2H=1/3(5^2)(2), which is approximately 52.4 miles cubic miles Find the volume of thdestroyed cone. V=1/3(3.14)r^2H=1/3(1^2)(0.5), which is approximately 0.52 cubic miles Find the percent of the original volcano removed by the eruption. (0.52/2.4)100, which is approximately 1%
 * 7.5 Surface Area and Volume of Cones
 * 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.
 * 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.
 * Oblique Cone- If Not,** The it is an oblique 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.
 * Net for a cone -
 * Solution:**The circumference of the base is C=2(3.14)r=14(3.14)
 * SA of a Right Cone-** The Surface Area, S, of a right cone with a lateral area L, base of area B, radius r, and slant height //L// is: S=L+B or S=(3.14)r//L//+(3.14)r^2
 * Volume of a Cone-** The Volume, V, of a cone with radias r, height H, and base area B is: V=1/3BH or V=1/3(3.14)r^2H
 * Example 2- A volcanologist is studying a violent eruption of a cone-shaped volcano. The original volcanic conehad a radius of 5 miles and a height of 2 miles. The eruption removed a cone-shaped area from the top of the volcano. this cone had a radius of 1 mile and a height 1/2 mile. What percentof the total volume of the original volcano was removed by the eruption?**
 * Solution:** Find the volume of the original volcano.

Objectives-** Define and use a formula for the SA and V of a sphere. Volume of a Sphere-The Volume, V, of a sphere whith radius r is: V=4/3(3.14)r^3 S=4(3.14)r^2=4(3.14)(27)^2=4(729)(3.14)=2916(3.14), which is approximately 9160.9 square feet Now multiply the surface area of 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, which is approximately $12,000
 * 7.6 Surface Area and Volume of Spheres
 * Sphere-** A sphere is the set of all points in soace that are the same distance,r, from a given point known as the center of the sphere.
 * Annalus-** When a plane intersects a sphere or a cylinder, it creates a ring-like cross section called the annalus.
 * Example 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=4/3(3.14)r^3=(3.14)(27)^3=4/3(19,683)(3.14)=26,244(3.14), which is approximately 82,488 cubic feet
 * SA of a Sphere-** The Surface Area, S, of a sphere with the radius r is: S=4(3.14)r^2
 * Example 2- The envelope of a hot-air balloon is 54 feet in diameter when inflated. The cost of the faberic to make the envelope is $1.31 per square foot. Estimate the total cost of the fabric for the balloon envelope.**
 * Solution:** First estimate the surface area of the inflated balloon envelope. The balloonis approximately a sphere with a diameter of 54 feet, so the radius is 27 feet.