A rectangular beam will be cut from a cylindrical log of radius r =30 inches. Suppose that the strength of a rectangular beam is proportional to the product of its width and the square of its depth. Find the dimensions of the strongest beam that can be cut from the cylindrical log.
s = kwd^2
since w^2 + d^2 = 60^2,
d^2 = 3600-w^2
s = kw(3600-w^2)
ds/dw = k(3600-3w^2) = 3k(1200-w^2)
ds/dw = 0 when w=20√3
so d = 20√6
To find the dimensions of the strongest beam that can be cut from the cylindrical log, we need to maximize the strength of the beam, which is proportional to the product of its width and the square of its depth.
Let's assume the width of the beam is w inches, and the depth is d inches.
The first thing we need to do is express the width in terms of the depth, as we are given a cylindrical log with a fixed radius.
The rectangle's width will be twice the distance from the center of the cylinder to the edge of the cylinder, which is equal to r.
So, the width of the beam (w) will be equal to 2r, or 2 * 30 = 60 inches.
Now, we can express the strength of the beam in terms of the depth (d) only:
Strength = width * depth^2 = 60 * d^2
To find the dimensions of the strongest beam, we need to find the maximum value of the strength.
To do this, we can take the derivative of the strength function with respect to the depth (d), set it equal to zero, and solve for d:
dStrength / dd = 0
Differentiating the strength function with respect to the depth (d) yields:
dStrength / dd = 2 * 60 * d = 120d
Setting this equal to zero:
120d = 0
Solve for d:
d = 0
Since the depth of the beam cannot be zero, this is not a valid solution.
Next, we need to consider the endpoints of the possible range of the depth.
Since we are cutting the beam from a cylindrical log with a radius of 30 inches, the maximum depth of the beam cannot exceed 30 inches.
Therefore, the endpoint is d = 30.
Now, we evaluate the strength of the beam at the critical points and endpoints:
For d = 0, we found this to be an invalid solution.
For d = 30:
Strength = 60 * 30^2 = 54,000.
So, the maximum strength of the beam is 54,000 and occurs when the depth (d) is equal to 30 inches.
Therefore, the dimensions of the strongest beam that can be cut from the cylindrical log are:
Width = 60 inches
Depth = 30 inches