# The strength, S, of a rectangular wooden beam is proportional to its width times the square of its depth. Find the dimensions of the strongest beam that can be cut from a 12 inch diameter cylindrical log.

## let the width of the beam be 2w and the depth be 2d

S = k(2w)(4d^2)
= 8k w(d^2)

I drew a circle and incribed a rectangle with the defined dimensions.
Notice the diagonal is 12
Construct a righ-angled triangle, with sides w, d, and hypotenuse 6
w^2 + d^2 = 36
d^2 = 36-w^2

in S = 8k w(d^2)
= 8k w(36-w^2)
= 8k (36w - w^3)

dS/dw = 8k(36-3w^2) , remember k is a constant
= 0 for a max of S
3w^2 - 36 = 0
w^2 = 12
w = √12 = 2√3
d^2 = 36-12 = 24
d = 2√6

so the width is 4√3 and the depth is 4√6

## Ah, the quest for the strongest beam! Let's dive into it, shall we?

To find the dimensions of the strongest beam, we need to maximize the strength, which is proportional to the width times the square of the depth. So let's call the width "w" and the depth "d".

Now, we have a cylindrical log with a 12-inch diameter. The diameter is essentially the width of the cylindrical log, so we know that w = 12 inches.

To determine the depth, we need to make use of the fact that the log is cylindrical. The depth of the beam we cut must be less than or equal to the height of the log, which is twice the radius.

Since the diameter is 12 inches, the radius is 12/2 = 6 inches. Therefore, the height of the cylindrical log is 2 * 6 = 12 inches.

Now, we can set up the equation for the strength of the beam:

S = w * d^2

Plugging in the values we know:

S = 12 * d^2

To find the dimensions of the strongest beam, we need to maximize S. This occurs when d is as large as possible, given the constraint that d must be less than or equal to the height of the log, which is 12 inches.

Hence, the dimensions of the strongest beam that can be cut from the cylindrical log are:
- Width (w): 12 inches
- Depth (d): 12 inches

Just remember, this is a theoretical calculation, and practical considerations such as the structural integrity of your log may come into play. Good luck and happy woodworking!

## 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.