MATHalino

Problem 817
Find the moment under the support R₂ of the continuous beam shown in Fig. P-817.
 

Problem 816
Determine the lengths of the overhangs in Fig. P-816 so that the moments over the supports will be equal
 

Problem 815
Find the moment at R₂ and R₃ of the continuous beam shown in Fig. P-815.
 

Problem 12
A circular sector of radius 10 cm is inscribed in a square of sides 10 cm such that the center of the circle is at the midpoint of one side of the square. Find the area of the sector in cm².
 

Problem 42
From a cylindrical glass 6 in. high and 3 in. in diameter, water is poured by tilting the glass until the highest point of the bottom of the glass lies in the plane of the water surface. How much water remains?
 

Problem 41
Soap kettles used in the commercial manufacture of soap are as a rule large cylindrical vats, 50,000 lb. or more of soap being made in a single beating. Find the capacity of such a kettle having an inside diameter of 18 ft and an altitude of 30 ft. if soap weighs 70 lb. per cu. ft.
 

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.
 

Problem 39
A grain elevator in the form of a frustum of a right circular cone is 24 ft. high, and the radii of its bases are 10 ft. and 5 ft., respectively; how many bushels of wheat will it hold if 1-1/4 cu. ft. equals 1 bu.?
 

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?

Problem 37
A factory chimney is in the form of frustum of regular square pyramid. The chimney is 125 ft. high the edges of its bases are 12 ft. and 8 ft., respectively. The cross-section of the flue is 6 ft. square. How many cubic feet of material does the chimney contain?