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?

Question

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?

Answer 1

#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)

Answer 2

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.

Answer 3

To design the strongest rectangular beams from logs with a radius of 72 cm, we need to find the dimensions that maximize the strength of the beam.

First, let's determine the formula for the strength of a rectangular beam. According to the given information, the strength of a rectangular beam is jointly proportional to its breadth (b) and the square of its depth (d^2).

Strength ∝ b * d^2

Next, we need to consider the constraints imposed by the log with a radius of 72 cm. Since the log is in the shape of a right circular cylinder, when we cut beams from it, the breadth of the beam will be twice the radius (2 * 72 cm) since it goes from one end of the cylinder to the other. The depth of the beam will be limited to the diameter of the cylinder, which is twice the radius (2 * 72 cm).

Therefore, the dimensions of the beam should be:

Breadth (b) = 2 * Radius = 2 * 72 cm = 144 cm
Depth (d) = Diameter = 2 * Radius = 2 * 72 cm = 144 cm

So, the dimensions of the beam that would produce the strongest rectangular beams from logs with a radius of 72 cm are a breadth of 144 cm and a depth of 144 cm.