I need help with this question:
The strength of a beam with a rectangular cross section varies directly as x and as the square of y. What are the dimensions of the strongest beam that can be sawed out of a round log with diameter d?
What am I supposed to do?
Thx in advance_
Assume the log to have a perfect circular section, of diameter d.
The radius is therefore r=d/2.
We have a choice of cutting a beam out of the log of width w, and height h, as long as sqrt(w²+h²)≤d.
We can eliminate "h" at the source using equality and the above Pythagoras relation, i.e.
h=sqrt(d²-w²)
Let the strength of the resulting rectangular beam be
S(w)=k*w*h²
=k*w*(sqrt(d²-w²)²
=k*w*(d²-w²)
where k is a constant of proportionality.
We look for the maximum value of S(w) by varying w, so we set dS/dw=0:
dS/dw=d(k(wd²-w³))/dw
=k(d²-3w²)
Equating dS/dw=0 and solving for w:
w=sqrt(d²/3)
and therefore
h=sqrt(d²-w²)
=sqrt(2d²/3)
To find the dimensions of the strongest beam that can be sawed out of a round log with diameter d, we need to determine the relationship between the beam's strength and its dimensions. According to the question, the strength of the beam depends on two variables, x and y. The relationship is given as:
Strength ∝ x⋅y^2
To find the dimensions of the strongest beam, we need to maximize this expression. However, we have another constraint, which is that the beam should be sawed out of a round log with diameter d. From the given information, we can assume that x and y are the width and height of the rectangular cross section of the beam, respectively.
To proceed, we need to relate x and y to the diameter of the log. Considering a rectangle inscribed in a circle, we can see that the diagonal of the rectangle is equal to the diameter of the circle. In this case, the diagonal corresponds to the hypotenuse of a right triangle formed by x, y, and the diagonal. So, using the Pythagorean theorem, we have:
x^2 + y^2 = d^2
Now we have two equations:
1. Strength ∝ x⋅y^2
2. x^2 + y^2 = d^2
We can solve these equations simultaneously to find the dimensions of the strongest beam that maximizes the strength.
1. Differentiate the strength function with respect to x and y to find the critical points where the maximum occurs.
2. Solve the system of equations composed of the derivative equations (obtained from the previous step) and the constraint equation (x^2 + y^2 = d^2).
3. Substitute the values of x and y obtained from the previous step back into the strength function to get the maximum strength.
This will give you the dimensions of the strongest beam that can be sawed out of a log with diameter d.