Flexure formula do not apply directly to composite beams because it was based on the assumption that the beam was homogeneous. It is therefore necessary to transform the composite material into equivalent homogeneous section. To do this, consider a steel and wood section to be firmly bolted together so that they can act as one unit. Shown below are the composite wood and steel section and the corresponding equivalent in wood and steel sections.
 

The quantity n is usually taken as the ratio of the moduli of elasticity of stronger material to the weaker material. In the above case, n = E_s / E_w.
 

Problem
The diameter of a sphere is 18 in. Find the largest volume of regular pyramid of altitude 15 in. that can be cut from the sphere if the pyramid is (a) square, (b) pentagonal, (c) hexagonal, and (d) octagonal.
 

Problem
A closed cylindrical tank measures 12 ft. long and 5 ft. in diameter. It has to contain water to a depth of 3 ft when lying in the horizontal position. Find the depth of water when it is in vertical position.
 

Problem 1
A circle is tangent to the line 2x - y + 1 = 0 at the point (2, 5) and the center is on the line x + y = 9. Find the equation of the circle.
 

Definition of circle
The locus of point that moves such that its distance from a fixed point called the center is constant. The constant distance is called the radius, r of the circle.
 

Problem 31
One section of the mainline sewer pipe for the city of Annapolis passes through a tunnel 484 ft. long. The cross section of the tunnel is shown in the figure. The sewer pipe is 16 in. in external diameter and is partially embedded, in a mattress of concrete, a cross section of which is shown in the figure. Find the total cost (a) of digging the tunnel at \$5.00 per cubic yard, (b) of pouring of concrete at \$9.00 per cubic yard.
 

Problem 30
Window glass formerly was made by gathering a lump of molten glass on the end of a hollow rod and blowing it into the form of large hollow cylinder about 6 ft. long and 1-1/2 ft. in diameter. This was cut longitudinally and then placed in an oven and heated until it softened, when it was flattened out into plates and cut into desired sizes. If this plate was 0.12 in. thick, find the amount of glass in the original lump.
 

Problem 29
A certain Chinese coin, 1/2 in. in diameter and 1/16 in. thick, is pierced by a square hole 1/8 in. on a side. Find the amount of metal in a coin.
 

Problem 28
A water tank, open at the top, consists of a right circular cylinder and a right circular cone, as shown. If the altitude of the cylinder is three times its radius, and the altitude of the cone is two times the same radius, find the number of square feet of sheet metal required to construct a tank having a capacity of 10,000 gal. (One gal. = 231 cu. in.)
 

Problem 27
One hundred and fifty posts are used in fencing a lawn. Each post is built in the form of frustum of a pyramid surmounted by a pyramid whose lower base is common with the upper base of the frustum. The height of the pyramidal top is 2 in. and the common base is a square 4 in. on an edge. The lower base of the frustum has an edge of 6 in. If the overall height of each is 6 ft., how much concrete will be used in making the posts?