The strength of a beam with rectangular corss-section is directly proportional to the product of the width and the square of the depth (thickness from the top to bottom of the beam). Find the shape of the strongest beam that can be cut from a cylindrical log of diameter d= 3 ft

To find the shape of the strongest beam that can be cut from a cylindrical log of diameter d=3 ft, we need to maximize the strength of the beam as per the given condition.

Let's assume the rectangular cross-section of the beam has a width W and a depth D. According to the given condition, the strength of the beam is directly proportional to the product of W and the square of D.

Strength ∝ W * D^2

To maximize the strength, we need to optimize the values of W and D. Since we are cutting the beam from a cylindrical log, the width W cannot be larger than the diameter of the log (3 ft). So, we can let W = 3 ft.

Now, we need to find the optimal value for the depth D. Since D represents the thickness from the top to bottom of the beam, it cannot be greater than the radius of the log. The radius of the log is d/2 = 3/2 = 1.5 ft.

Therefore, the shape of the strongest beam that can be cut from the cylindrical log is a rectangular beam with a width of 3 ft and a depth of 1.5 ft.