040 Review Problem - Volume comparisons of two water crystals

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Problem 40
When a liquid freezes, it changes into a mass of solid bodies, each of which has a definite geometric form and is known as a crystal. Crystals are always bounded by plane surfaces, which are arranged in an orderly fashion with reference to imaginary lines drawn through the crystal and called its area. Every crystal has therefore a definite geometric form. Compare the volumes of crystals, one in the shape of a hexagonal right prism 'a' units on a side and '2a' units in altitude, and the other in the shape of two oppositely directed hexagonal regular pyramids, each 'a' units high with a common base 'a' units on a side.



038 Review Problem - Circular log with non-uniform cross-section

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Problem 38
A log 18 ft. long is 2 ft. in diameter at the top end and 3 ft. in diameter at the butt end.



  1. How many cubic feet of wood does the log contain?
  2. How many cubic feet are there in the largest piece of timber of square cross section that can be cut from the log?
  3. How many cubic feet are in the largest piece of square timber of the same size throughout its whole length?
  4. How many board feet does the piece of timber in (c), a board foot being equivalent to a board 1 ft. square and 1 in. thick?

Hint: In (b) the larger end is the square ABCD. What is the smaller end? In (c) one end is the square EFGH. What is the other end?

035 Review Problem - Amount of concrete in a caisson

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Problem 35
In the foundation work of the Woolworth Building, a 55-story building in New York City, it was necessary, in order to reach the bedrock, to penetrate the sand and quicksand to a depth, in some instances, of 131 ft. If the largest circular caisson, 19 ft. in diameter, was 130 ft. deep and was filled with concrete to within 30 ft, of the surface, how many cubic yards of concrete were required?