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.
Let's assume that the beam has a rectangular cross-section with width w and depth d. According to the given information, the strength of the beam is proportional to the product of the width and the square of the depth, i.e., S ∝ w * d^2.
Since the log is cylindrical, the width of the beam must be less than or equal to the diameter of the log, w ≤ d.
To maximize the strength S, we need to maximize the product w * d^2. This can be achieved by choosing the maximum values for both w and d.
Since we need to maximize w * d^2, let's consider the maximum possible values for both w and d.
Since the width w should be ≤d, the maximum possible value for w is d.
So, w = d = 3 ft. Therefore, the width of the beam should also be 3 ft.
To find the maximum value for the depth d, let's look at the cylindrical log. The diameter of the log is given as 3 ft.
Since the depth d is the thickness from the top to the bottom of the beam, we need to find the maximum thickness that can be cut from the log.
The maximum thickness that can be cut from the log is half of the diameter, which is d/2 = 3/2 = 1.5 ft.
Therefore, the shape of the strongest beam that can be cut from the cylindrical log of diameter d = 3 ft is a rectangular beam with a width of 3 ft and a depth of 1.5 ft.
To find the shape of the strongest beam that can be cut from a cylindrical log, we need to determine the dimensions of the beam that will maximize the strength.
Given that the strength of a beam with rectangular cross-section is directly proportional to the product of the width and the square of the depth, we can denote the width as "w" and the depth as "d".
Let's assume that the width and depth of the rectangular beam are equal, which means the beam is a square. Therefore, w = d.
To find the shape of the strongest beam, we can consider the area of the cross-section. The area (A) of a square cross-section is given by A = w * d.
Since we assumed w = d, we can say A = w * w, which simplifies to A = w^2.
To maximize the area, we need to maximize the value of w^2. This occurs when w is maximized.
Now, let's consider the cylindrical log. We know that the diameter (d) of the log is given as 3 feet. The width (w) of the square beam cannot exceed the diameter (d), so w ≤ d.
Therefore, to maximize w (and consequently the area and strength of the beam), we need to set w equal to the diameter of the cylindrical log. Hence, the shape of the strongest beam that can be cut from the log is a square cross-section with a width and depth of 3 feet.
Strength, S = xy²
But we know that
x²+y²≤(3/2)²
therefore, substitute
S(x)=x((3/2)²-x²)
For maximum strength,
dS(x)/dx = 9/4-3*x^2 =0
=>
x²=3/4
x=sqrt(3/4)
y^2=(3/2)^2-x^2=9/4-3/4=6/4
y=sqrt(1.5)