It is useful to know that the lengths of lines required in setting out the pattern for any conical article may be obtained by calculation, without previously having drawn an elevation of the object. An example of this method is shown by the calculations for the pattern in Fig. 89. Suppose the article to have the following dimensions: Top 9 in. diameter, bottom 2 ft. diameter, and depth 1 ft. 6 in.; then the height of complete cone, of which the vessel is a part, can be calculated as follows: -
Height = C D = radius of base x depth / radius of base - radius of top
=12 X 18 / 7.5 = 28.8 in.
And slant height of cone - that is, radius of outer curve of pattern can be found by the use of the property of the right-angle triangle, thus-
C B = = 31.2 in.
The radius for the inner curve of pattern can be found as above, or from the following rule: -
C A = radius of top x slant height of complete cone / radius of base
= 4.5 X 31.2 / 12 = 11.7 in.
The length round outer curve of pattern will, of course, be-
24 x 3.1416 = 75.4 in., so that it will be seen that the complete pattern of the article, whether it is made up in one or more pieces, can be set out from calculated dimensions. This method is especially useful in large work, or where a high degree of accuracy is required.
Capacity of a Conical Vessel.
Seeing that we have the above calculations before us, it will be as well to go over what is usually considered to be a somewhat difficult task - that is, to find the capacity of a circular tapering vessel. The simplest plan to adopt is to calculate the volume of the complete cone, and then to deduct from it the volume of the small cone, which we can imagine is cut off the top.
The rule for finding the volume of a cone is:- "Multiply the area of the base by one-third the vertical height." From the previous calculations in reference to Fig. 89, we have both the height of complete cone and of the small cones cut away. The volume of the vessel then is-
(12)2 ╥ X 28.8 / 3 - (4.5)2╥ X 10.8 / 3
= 144 ╥ x 28.8 / 3 - 20.25 ╥ x 10.8/ 3
= 1,309 ╥ = 1,309 x 3.1416 = 4, 114 c.in.
The number of cubic inches in an imperial gallon is 277.274. This is a most awkward number to use; but as it has been fixed by Act of Parliament (in 1826) as the volume of a gallon, we have to make the best of it. The capacity of the above vessel will therefore be -
4,114 / 277.274 = 14.84 gal.
The capacity may also be calculated, but not quite so accurately, by remembering that a cubic foot contains as near as possible 6¼ gal. Thus -
4114 / 1,728 x 6¼ = 14 7/8 gal. (nearly).
As a gallon of fresh water is usually taken to weigh 10 lb., the weight of water in the vessel will be -
14.84 x 10= 148.4 = 148½ lb. (nearly).