Machinery's Handbook, 31st Edition
SPHERE FORMULAS
57
2
π 5 2 20 2 4 + ---- =
2 E 4 + ---
392.70 cm 2
S a π F =
=
S v π F E 2 8 =
2
π 5 20 2 8 =
2
--- F 6 + ---
850.85 cm 3
---- 5
6 + ---
=
× Example 4: Find the outside and inside surface area U a and volume U v of section U of a sphere if R 1 = 5.0 inches, R 2 = 4.0 inches, and included angle f = 30 ° . Solution: Sectional area U a and sectional volume U v : U a 2 π R 1 2 R 2 2 + ( ) 1 2 π 5 2 4 2 + ( ) 1 30 ° 2 – cos ---- 8.78 in 2 = = = Example 5: Find the total surface area W a and volume W v of ring W, if R 1 = 5.0 inches and R 2 = 4.0 inches. Solution: Sectional area W a and sectional volume W v : W a 4 π 2 R 1 R 2 4 π 2 5 4 × × 789.57 in 2 = = = W v 2 π 2 R 1 R 2 2 2 π 2 5 4 2 × × 1579.14 in 3 = = = φ 2 – cos -- U v 2 π R 1 = 3 R 2 3 – ( ) 1 φ 2 – cos -- 2 π 5 3 4 3 – ( ) 1 30 ° 2 – cos ---- 13.06 in 3 = = Parabola.— A parabola is the set of all points P in the plane that are equidistant from a focus F and a line called the directrix. The parts of the parabola are labeled in the figure below. The general equation of a parabola horizontal axis is given by y k – ( ) 2 4 p x h – ( ) = , where the vertex is located at point ( h, k ), the focus F at point ( h + p , k ), and the directrix is the line x -axis at x = h - p . The latus rectum is the vertical line segment through the focus; its endpoints are on the parabola. Hence, it lies on the line x = h + p . The length of the latus rectum is four times the absolute value of the x -coordinate of the focus, hence | 4( h + p ) | . Example: Determine the focus, directrix, parabolic axis, vertex, and length of the latus rectum of the parabola 4 y 2 8 x – 12 y – +1 = 0 Solution: Rewrite the equation in the general form of a parabolic equation (see Solving by Completing the Square on page 34).
4 y 2 8 x – 12 y – +1 = 0 4 y 2 12 y – 8 x = – 1 y 2 3 y – 2 x 1 4 = – --
2
y 2 3 y 3
2 x 1 4
2 – -- 3 2 -- +
– -- 9 4 + --
=
2
y 3 2 – --
2 x +1 ( ) =
Parabola
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