You work as a structural engineer in a multinational construction company which also produces building materials such as beams from logs in the shape of right circular cylinders. You know that the strength of a rectangular beam is jointly proportional to its breadth and the square of its depth. You are tasked to design the strongest rectangular beams from logs whose radius is 72 cm. What must be the dimensions of the beam that you would produce?
2 answers
The company you work in makes decorative grills for doors and windows of big condominium buildings. The company received orders for steel designs of circle and square figures to be used in window grills of a four-story condominium . As a materials estimator, you are to design these figures using stainless steel rods 100 cm long to be cut into two pieces and bent into the desired figures. How should the rods be cut so that the combined area of the two figures is as small as possible? Make samples of the figures using only iron wire.
#1. s = kbd^2
I don't see anything about the length of the beam. How are the breadth and depth related? We don't generally consider circles as having breadth and depth.
#2. Let the square have side s, and the circle have radius r. So,
4s + 2πr = 100
r = (50-2s)/π
That means the area is
a = s^2 + πr^2 = s^2 + (50-2s)^2/π = 1/π (πs^2 + (50-2s)^2)
da/ds = 2((π+4)s - 100)
so minimum area happens when s = 100/(π+4)
I don't see anything about the length of the beam. How are the breadth and depth related? We don't generally consider circles as having breadth and depth.
#2. Let the square have side s, and the circle have radius r. So,
4s + 2πr = 100
r = (50-2s)/π
That means the area is
a = s^2 + πr^2 = s^2 + (50-2s)^2/π = 1/π (πs^2 + (50-2s)^2)
da/ds = 2((π+4)s - 100)
so minimum area happens when s = 100/(π+4)