## To find the dimensions of the strongest beam that can be cut from a cylindrical log, we need to maximize the strength of the beam.

Let's assume that the width and depth of the beam are both equal to "x" inches.

First, we need to express the diameter of the cylindrical log in terms of "x". The diameter of the cylindrical log, which is equal to twice its radius, is 12 inches. So, the radius is 12/2 = 6 inches.

Next, let's find the relationship between the width and depth of the beam. The problem states that the strength, S, of the beam is proportional to its width times the square of its depth. Mathematically, this can be expressed as:

S = k * width * depth^2

where k is a constant of proportionality. Since we are looking for the maximum strength, we can ignore the constant k.

Now, we can substitute the width and depth values in terms of "x" into the equation:

S = x * x^2 = x^3

To find the dimensions of the strongest beam, we need to find the value of "x" that maximizes the strength. Since "x" represents both the width and depth, it needs to be positive and less than or equal to half the diameter of the cylindrical log.

In this case, "x" must be less than or equal to 6 inches. However, since the problem is asking for the strongest beam, we need to find the maximum value of "x" that satisfies this condition.

To do this, we take the derivative of the strength equation with respect to "x" and set it equal to zero to find the critical point:

dS/dx = 3x^2 = 0

Solving this equation, we find that x = 0 or x = 0, which doesn't give us any useful information since the width and depth of the beam cannot be zero.

Therefore, the only value of "x" that satisfies the condition is x = 6 inches.

Hence, the dimensions of the strongest beam that can be cut from a 12-inch diameter cylindrical log are a width and depth of 6 inches each.