All truths are easy to understand once they are discovered; the point is to discover them. Galileo

Transcription

1 Section 7. olume All truts are easy to understand once tey are discovered; te point is to discover tem. Galileo Te main topic of tis section is volume. You will specifically look at ow to find te volume of various tree dimension geometric objects suc as rectangular solids and cylinders. Te volume of an object is te amount space occupied by te object. Te procedure for finding volume of an object is similar to finding te area of an object in tat it is simply a matter of substituting correct values into te proper formula. Here is a list of some of te key formulas used to find volume. Geometric Sape Sketc Formula Rectangular Solid = lw Cylinder = π r Cube = s Now, let s find te volume of some tree dimensional objects.

2 Example 1 Find te volume of a cylinder wit a radius of inces and eigt of 4 inces. in 4 in Solution: Substitute te values of te radius and te eigt into te volume formula. = π r = π (cm) = π (9 cm = 6π cm (4 cm) )(4 cm) Example Find te volume of rectangular solid tat is feet by feet by 4 feet. 4 feet feet feet Solution: Substitute te values of te lengt, widt, and eigt into te volume of rectangular solid formula. = l w = = ( ( (4 = (6 ft )(4 4 ft

3 Example Suppose you ave a cylinder saped ot water eater tat as a eigt of 5 feet and a radius of 1 foot. How muc water can te ot water eater old? 1 foot 5 feet To find te answer, substitute te values of te radius and te eigt into te volume of a cylinder formula. Notice tat te units of te final answer are in cubic feet. = π r = π (1 = π (1cm = 4π ft 1.56 ft (4 )(4 Example 4 A fis aquarium saped like a rectangular solid is 0 inces wide, 0 inces long, and 0 inces tall. How muc volume could te fis aquarium old? Since te fis aquarium is a rectangular solid, just substitute te lengt, widt, and eigt of te aquarium into formula of te volume of a rectangular solid. Te final units will be in cubic inces. 0 inces 0 inces 0 inces = l w = ( 0in)(0in)(0in) = (600in )(0 = 1000 ft Te fis aquarium will be able to old 1000 cubic feet of water

4 Surface Area Every tree dimensional object as bot volume and surface area. Te volume, as stated earlier, measured te amount of space occupied by a tree dimension object. Te surface area of a tree dimension object measures amount of surface of te object. Te surface area of object suc as a cube, rectangular solid, pyramid, or cylinder is found by find te area of eac face and ten finding te sum of te faces. For example, suppose you ad to find te surface area of a rectangular solid. You would ave to find te area of top face, bottom face, front face, te face in te back, and te two sides. Let s try an example. Example 5 Find te surface area of a rectangular tat as a lengt of 6 feet, eigt of 4 feet, and a widt of 5 feet. Te faces of te rectangular solid are all rectangles. Terefore, you can use te formula for te area of a rectangle to find te area of eac face. A = l w 4 feet 5 feet 6 feet. To find te surface area of te object, you find te area of eac face and ten sum up all of te faces. Te faces in te front and back of te rectangular solid are te same so use te area of a rectangular to find te area. In tis part you will use te lengt of 6 feet and eigt of 4 feet as te dimensions of te rectangle. A = l = ( 6 (4 = 4 ft Te top and bottom of te rectangular solid are also te same. Tese areas can be found by multiplying te lengt by te widt. A = l w = ( 6 (5 = 0 ft Te two sides of te rectangular solid are also te same. Tese areas can be found by multiplying te widt by te eigt. A = w = ( 5 (4 = 0 ft

5 Te total surface area of te rectangular solid can be found by taking te sum of all te sides of te objects. Remember to multiply te area of eac rectangle found by two since te rectangles occur in pairs. A = = (4 ft ) + (0 ft ) + (0 ft ) = 48 ft + 60 ft + 40 ft 148 ft You also can use te following formula to find te surface area of a rectangular solid. A = l w + l + w Substituting te values for lengt, widt, and eigt form te rectangular solid, you will get te same value for te surface area. A = = (6 (5 + (6 (4 + (5 (6 = 60 ft + 48 ft + 40 ft 148 ft Example 6 Find te surface area of a cylinder in example 1. Recall tat te cylinder ad a eigt of 4 inces and a radius of inces. in 4 in To find te surface area of a cylinder, you must again find te area of eac face. Terefore, you must find te area of te top, bottom, and te rectangle tat wraps around te object. in l = πr 4 in in

6 First find te area of te circle at te top and bottom of te cylinder. A = ( in) 9π in π r = π = Next, find te area of te area of te middle section wic is a rectangle. Te lengt of te rectangular is te same as te circumference of te circle and te widt of te rectangle is te eigt of te cylinder. Terefore, te area of te rectangle can be found te following formula. lengt = circumference of circle = π r Area of rectangle = l w = πr Tus te area of te rectangle is A = πr = π ( in)(4 in) = 4π in Terefore, te total surface area can be found by taking te sum of te faces. A = (9π in ) + 4π in = 18π in + 4π in = 4π in Example 7 Suppose you want to mail a rectangular solid saped package tat measures 0 cm by 15 cm by 1 cm. How muc postal wrap do you need to completely cover te package? If te postal rate is.1 cents per cubic centimeter, find te cost to mail te package. To find te amount of postal wrap to cover te package, you need to find te surface area of te package. A = l w + l + w A = (0 cm)(15 cm) + (0 cm)(1 cm) + (15 cm)(1 cm) A = 600 cm A = 1440 cm cm + 60 cm To find te cost to mail te package, you must multiple te surface area of te object by te postal rate. cents Cost =.1 (1440cm ) = 140 cents or $1.44 cm

7 Example 8 A soup can as a radius of 4 cm and a eigt of 10 cm, ow muc paper in square centimeters would be needed to make a label tat would fit around te soup can. 4 cm 10 cm Te label would be a rectangle tat would ave a eigt of 10 cm and a lengt tat would be te same as te circumference of te base. Lengt = π r = π (4 cm) = 8π 8 π cm 10 cm Use te area of rectangle you will get tat te area is 80 π or 15. cm Area = l w = ( 8π m cm)(10 cm) = 80π cm Mat History Excursion: Truncated Pyramids As discussed before in tis capter, te Egyptians did not make use of formulas or variables. Te Egyptians did owever look at specific examples involving te volume of a truncated pyramid. Tis example comes from problem 14 of te Moscow Papyrus wic now resides in te Museum of Fine Arts in Moscow, Russia. A truncated pyramid is basically a regular pyramid wit its top removed as sown in te following figure.

8 Te calculations te Egyptians use to find te volume of a truncated pyramid translated to te following formula. ( a + ab b ) = + In tis formula a represents te lengt of te smaller base at te top, b represents te lengt of te larger base at te bottom, and is te eigt of te truncated pyramid. Here is an example ow tis formula can be used find te volume of a truncated pyramid. Example 9 (Problem 14 Moscow Papyrus) Find te volume of a truncated pyramid wit a lower base wit a lengt 4 cubits, an upper base wit a lengt cubits, and a eigt of 6 cubits. To find te volume of te truncated pyramid, you will substitute cubits in for a, 4 cubits in for b, and 6 cubits in for into te truncated pyramid formula. cubits cubits 6 cubits 4 cubits = 6 = = 4 cubits ( a + ab + b ) ( + (4) + 4 ) ( ) = (8) = 56 cubic cubits

9 Te Regular Pyramid Wen we tink of a pyramid we usually tink of a regular pyramid wit a square base. Te pyramids in Egypt are regular pyramids wit a square base Here is te formula to find te volume of a pyramid wit a square base. b 1 = b b

10 Example 10 Find te volume of a truncated pyramid wit a eigt of 0 cubits, an upper base of a = 6 cubits, and a lower base of b = 4cubits. After you find te volume of te truncated pyramid, find te volume of a regular pyramid wit a base of 4 cubits and a eigt of 0 cubits. Compare te results of te two pyramids. Truncated Pyramid = 0 cubits a = 6 cubits b = 4 cubits a = 6 cubits = 0 cubits b = 4 cubits 0 ( b + ab + a ) = ( 4 + 4(6) + 6 ) = 10( ) = 10(765) 7560 cubits = = Regular Pyramid = 0 cubits b= 4 cubits (576)(0) = b = (4) (0) = = 5760 cubits Compare te two volumes cubic cubits 5760 cubic cubits = 1800 cubic cubits Te truncated pyramid as a volume of 1800 more cubic cubits.

11 Exercises 1) Find te volume of a cylinder wit a radius of inces and eigt of 4 inces. ) Suppose you ave a cylinder saped ot water eater tat as a eigt of 5 feet and a radius of 1 foot. How water can te ot water eater old? ) Find te volume of following rectangular solid. feet 7 feet 4 feet 4) A fis aquarium saped like a rectangular solid is 0 inces wide, 0 inces long, and 0 inces tall. How muc volume could te fis aquarium old? 5) A soup can as a radius of 5 cm and a eigt of 8 cm, ow muc paper in square centimeters would be needed to make a label tat would fit around te soup can. 6) Suppose you want to mail a rectangular solid saped package tat measures 10 cm by 1 cm by 1 cm. How postal wrap do you need to completely cover te package? If te postal rate is.1 cents per cubic centimeter, find te cost to mail te package. 7) Find te volume of a truncated pyramid wit a eigt of 0 cubits, an upper base of a = 4 cubits, and a lower base of b = 0cubits. 8) Find te volume of a regular pyramid wit a base of 1 cubits and a eigt of 14 cubits.