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Section 8.3 Nack/Jones1 8.3 Cylinders & Cones

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Section 8.3 Nack/Jones2 Cylinders A cylinder has 2 bases that are congruent circles lying on parallel planes. The cylinder is a right cylinder if the line joining the centers of the circles called an axis is an altitude—that is, perpendicular to the planes of the circular bases.

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Section 8.3 Nack/Jones3 Surface Area of a Cylinder Theorem 8.3.1: The lateral area L of a right circular cylinder with altitude of length h and circumference C of the base is given by: L = hC or L = 2 rh Theorem 8.3.2: The total area T of a right circular cylinder with base area B and lateral area L is given by: T = L + 2B or T = 2 rh + 2 r² Example: For the figure: L = 2 5 12 = 120 sq. units T = 120 + 2 5² = l70 sq. units. h = 12 5

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Section 8.3 Nack/Jones4 Volume of a Cylinder Theorem 8.3.3: The volume V of a right circular cylinder with base area B and altitude of length h is given by: V = Bh or V = r²h Example: The volume of the right cylinder with radius = 5 and height = 12 is: V = 5² (12) = 60 units 3

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Section 8.3 Nack/Jones5 Cone The solid figure formed by connecting a circle with a point not in the plane of the circle is called a cone. A cone has one base. Right cone: the altitude passes through the center of the base circle. Also the slant height is the distance l Oblique cone: If the altitude, h, is not perpendicular to the base.

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Section 8.3 Nack/Jones6 Surface Area of a Cone Theorem 8.3.4: The lateral area L of a right circular cone with slant height of length l and circumference C of the base is given by: L = ½ l C or L = ½ l (2 r) Theorem 8.3.5: The total area T of a right circular cone with base area B and lateral area L is given by: T = B + L or T = r² + r l Theorem 8.3.6: In a right circular cone, the lengths of the radius r (of the base), the altitude h, and the slant height l satisfy the Pythagorean Theorem; that is l ² = r² + h² in every right circular cone.

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Section 8.3 Nack/Jones7 Volume of a Cone Theorem 8.3.7: The volume V of a right circular cone with base area B and altitude of length h is given by: V = ⅓ Bh V = ⅓ r²h Solids of Revolution Locus of points when we rotate a plane region around a line segment which becomes the axis of the resulting solid formed. Figure 8.35 and Example 6 p. 415 - 6

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Section 8.3 Nack/Jones8 Solids of Revolution P. 394-5 Practice exercises: p. 397, #24.

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