Hi, I'm having trouble with this problem...
"If the strength of a rectangular beam of wood varies as its breadth and the square of its depth, find the dimensions of the strongest beam that can be cut out of a round log, diameter d. "
I can't come up with an equation with only 1 variable.
Help me, Please!!!
Many thanks, Stu.
Im not really sure but i think its
360(d)(120)
Hi Levy, any chance you could explain how you came up with that solution?
I believe I should find values for Breath and Depth with relation to diameter, but I am stuck for a way to do it.
Many thanks.
rectangle must fit in circle of diameter d
B =chord1 of circle = d sin (theta1/2)
H = chord 2 of circle = d sin( [pi-theta]/2)
because the angles subtended by the inscribed breadth and depth add up to a straight line. pi is straight line angle in radians.
S = k B H^2
S = k d sin (theta/2)*d^2 sin^2 ([pi-theta]/2)
let A = theta/2 to make it all easier
S = k d^3 sin A sin^2 (pi/2-A)
but sin (pi/20-A) = cos(A)
so
S = k d^3 sin(A) cos^2(A)
take derivative and set to 0
0 = dS/dA = k d^3[ -2 sin^2(A)cos(A)+ cos^3(A)]
2 sin^2 (A) = cos^2(A)
but cos^2(A) = 1 - sin^2(A)
2 sin^2 A = 1 - sin^2(A)
3 sin^2 (A) = 1
sin(A) = sqrt(3)/3 = about .577
so A is about 35.2 degrees
so
B = d sin (A) = .577 d
H = d sin(90-A) =d sin 54.8deg = .817 d
Damon, you're a God-send!
If you're ever in North Wales I'll buy you a pint!
1x10^3 Thanks.
Stu.
Hi Stu, I'd be happy to help you with this problem!
To find the dimensions of the strongest beam that can be cut out of a round log, we need to maximize the strength of the beam. Let's assume the breadth of the beam is "b" and the depth is "d". Now, we need to come up with an equation that relates these variables.
The problem tells us that the strength of the beam varies as its breadth and the square of its depth. This means that the strength (S) of the beam can be expressed as:
S = b * d^2
Since we're trying to find the dimensions of the strongest beam, we want to maximize S. To do that, we need to find the maximum value of the function S = b * d^2.
To find the maximum of a function, we typically take the derivative with respect to the variable we want to maximize (in this case, either b or d), set the derivative equal to zero, and solve for the variable. However, in this case, we have two variables in our equation.
To solve this problem, we can use the fact that the beam is cut from a round log with diameter d. This means that the breadth and depth of the beam cannot exceed d.
To find the maximum value of S, we need to find the maximum value of both b and d, while ensuring that b and d do not exceed d. In other words, we need to find the maximum value of b and d, subject to the constraint that b and d are less than or equal to d.
Now, let's break it down further:
1. We're looking for the maximum value of b and d within the constraints b <= d and b, d <= d. This means that both b and d must be less than or equal to d.
2. Since the problem asks for the dimensions of the strongest beam, we need to find the values of b and d that maximize S = b * d^2, subject to the constraints mentioned above.
3. To solve this, we consider two cases:
a) Case 1: Let's assume that b = d.
In this case, S = b * d^2 = d * d^2 = d^3.
As d is the maximum value of both b and d, S is also maximized at d.
b) Case 2: Let's assume that b < d.
In this case, let's consider what happens when we increase b:
- If b is increased, d * b^2 also increases, as long as it remains less than or equal to d.
- However, if b becomes greater than d, the condition b <= d is violated.
- Hence, the maximum value of b is d.
From the above analysis, we can conclude that in both cases, the highest value for S = b * d^2 is obtained when b = d.
Therefore, the dimensions of the strongest beam that can be cut out of a round log with diameter d are b = d and d = d.
Hope this helps! Let me know if you have any further questions.