For an oblique prism or cylinder, imagine starting with a right prism or cylinder and sliding thin layers to make it oblique. An oblique pyramid or cone has the same formula as a right pyramid or cone. Imagine starting with a right pyramid or cone and sliding thin layers to make it oblique. Jun 2 8:49 PM Do Now 1: Application of Cavalieri's Principle Find the volume of both prisms and explain how Cavalieri's Principle applies. A CD case measures 142mm by 125mm by 10mm. (BTW, there are 17 CD cases.) A base =142(125) =17,750 V=17,750(170) =3,017,500 V 1 CD =142(125)(10) 177,500 V=177,750(17) =3,017,500 Jun 2 9:02 PM 1
Do Now 2: Jenny says that the two prisms DO NOT have the same volume because the cross sections are not the same. Renee disagrees; she says that it isn t the shape that has to be the same it is the area. Renee thinks they have the same volume. Who is right and why? A base =1/2(12)(5) A base=(6)(5) =30 cm 2 =30 cm 2 Jun 2 9:02 PM Do Now 3: Jared wants to test out a new theory. Instead of having the cross area sections the same as Cavalieri suggested he wants to half the radius of one cross section and then double the height to make up for it. He believes because he divided the radius by 2 but doubled the height that the volumes should be equal. Is he correct? Explain. A slice =π(4 2 )=16π V=16π(5) = 80π cm 3 A slice =π(2 2 )=4π V=4π(10) =40π cm 3 The volumes are not the same. He would need to "quadruple" the height instead of just doubling it. Jun 2 9:04 PM 2
Exercises: Find the volumes: (Leave answers in simplest radical form or in terms of if applicable.) Oblique trapezoidal prism 4) 5) Oblique circular cylinder A base =(8+4)/2 (9) =54 V=54(15) = 810 cm 3 A base =6 2 π = 36π V=36π(15) = 540π un 3 Jun 2 9:06 PM Exercises: Find the volumes: (Leave answers in simplest radical form or in terms of if applicable.) Oblique rectangular prism 6) 7) Oblique circular cone 3 3 30, 60, 90 x, x 3, 2x 2x=6 x=3 A base =(3 3)(10)=30 3 V=30 3(8) = 240 3 cm 3 A base =3 2 π = 9π V=(1/3)9π(5) = 15π cm 3 In three dimensions, a parallelepiped is a prism whose faces are all parallelograms. It is unclear in #6 if the base is a rectangle or a parallelogram. Until you use the bottom as a base, and the volume is the same as using the front parallelogram as the base. But since a rectangle is a special parallelogram, I suppose we can argue that it is a parallelepiped afterall. Jun 2 9:08 PM 3
8) Find the volume if the height of the oblique prism is 12 mm and the triangle base has a base of 6 mm and a height of 4 mm. 9) Find the volume of the oblique pentagonal prism if a side length for the base is 7cm, the height of the prism is 24 cm and the apothem for the base is 4.8 cm. A base =(6)(4)/2 =12 V=12(12) = 144 mm 3 A base =1/2(7)(5)(4.8) =84 V=84(24)=2016 cm 3 Jun 2 9:08 PM *10. An enclosed glass box contains 1620 in 3 of water. When the glass box is tilted on its side the water shifts places. a) What is the relationship of the water before and after the tilting? The volume of the water does not change. V=27(12)(18) = 1620 in 3 Jun 2 9:15 PM 4
*10. An enclosed glass box contains 1620 in 3 of water. When the glass box is tilted on its side the water shifts places. b) What is the height of the water when he box is tilted upright? 12(18)(h) = 1620 216h = 1620 h = 7.5 in. Jun 2 9:15 PM Thanks, Sr. Cavalieri & Zu Geng Bonaventura Cavalieri(1598 1647) was an Italian mathematician. He was a precursor of infinitesimal calculus. Cavalieri, Kepler and other mathematicians, who lived during the century preceding Newton and Leibniz, invented and used intuative infinitesimal methods to solve area and volume problems. Twenty years later the publication of Kepler's Stereometria Doliorum, Cavalieri wrote a very popular book: Geometria indivisibilibus (1635). In this book, the Italian mathematician used what is now known as Cavalieri's Principle. Zu Geng, born about 450, was a Chinese mathematician who used what is now know as the Principle of Liu Hui and Zu Geng to calculate the volume of a sphere. Liu Zu theory is equivalent to Cavalieri's Principle. May 23 6:22 PM 5
May 23 6:32 PM https://schoolyourself.org/learn/geometry/cavalieri 2d http://math.tutorvista.com/geometry/cavalieris principle.html http://www.regentsprep.org/regents/math/geometry/gg2/prismpage.htm http://thinkzone.wlonk.com/area/areavol.htm http://www.mathopenref.com/cylinderoblique.html https://www.sophia.org/tutorials/volume of an oblique cone May 23 6:15 PM 6