Page:Sheet Metal Drafting.djvu/181

This page has been validated.

FRUSTUMS OF CONES

167

75. Related Mathematics on Short Handled Dipper.—Volume of a Frustum of a Cone of Revolution.—The frustum of a cone has a circular top and a circular base. These are known as the upper and lower bases of the frustum. The altitude of the frustum is the shortest distance between the upper and lower bases, and is always measured perpendicularly. The volume of a frustum is found by adding together the area of the upper base, the area of the lower base, and the square root of the product of the upper base area times the lower base area; the sum of these quantities is then multiplied by one-third of the altitude. Expressed as a formula

in which

{\begin{aligned}V&=(B+b+{\sqrt {(B\times b)}}\times H\div 3\\V&={\text{Volume}}\\B&={\text{Upper base area}}\\b&={\text{Lower base area}}\\H&={\text{Altitude}}\end{aligned}}

In applying this formula to Fig. 236, the areas of the upper and lower bases must first be computed.

Area of Circle	=	D²×.7854 (Chapter II)
6¼²×.7854	=	30.68 area of upper base
4½²×.7854	=	15.904 area of lower base

Known values can now be substituted in this formula

$\scriptstyle {V=(30.68+15.904+{\sqrt {30.68\times 15.904}})\times 3{\frac {3}{4}}\div 3}$

Performing the arithmetic:

\scriptstyle {3{\frac {3}{4}}\div 3={\stackrel {\scriptstyle {~5}}{\frac {{\cancel {1}}{\cancel {5}}}{4}}}\times {\frac {1}{\cancel {3}}}=1{\frac {1}{4}}}

15.904
30.68
127232
95424
477120
487.93472

4!√487.934722.08+

4|√4
42|087
42|084
4408|3 9347
4408!3 5264
4408 !34083