A wooden beam has a rectangular cross section of height h and width w (see figure).The strength S of the beam is directly proportional to the width and the square of the height.What are the dimensions of the strongest beam that can be cut from a round log of diameter d = 26 inches? Round your answers to two decimal places.[Hint: where k 0 is the proportionality constant.]
A) inches and inches
B) inches and inches
C) inches and inches
D) inches and inches
E) inches and inches

Question

A wooden beam has a rectangular cross section of height h and width w (see figure).The strength S of the beam is directly proportional to the width and the square of the height.What are the dimensions of the strongest beam that can be cut from a round log of diameter d = 26 inches? Round your answers to two decimal places.[Hint:  where k > 0 is the proportionality constant.]  ​
A) inches and inches 
B) inches and inches 
C) inches and inches 
D) inches and inches 
E) inches and inches

Quizplus · Accepted Answer

The area of the cross section of the beam is A = hw, and the radius of the log is r = d/2 = 13 inches. From the Pythagorean theorem, we have h^2 + w^2 = (2r)^2 = 676, so we can express the area A in terms of h only:
A = h(sqrt(676 - h^2))
The strength S is proportional to the width w and the square of the height h:
S = kw^2h^2
where k is a proportionality constant. We want to maximize S with respect to h, subject to the constraint A = hw. We can use Lagrange multipliers to solve this optimization problem.
Let L = S - λA, where λ is the Lagrange multiplier. Then
dL/dh = 2khw^2 - λ(sqrt(676 - h^2) + hh/sqrt(676 - h^2)) = 0
dL/dw = 2kw^2h = 0
dL/dλ = A - hw = 0
From the second equation, we see that either w = 0 or h = 0, which is not a valid solution. Therefore, h is nonzero and we can divide the first equation by h and solve for w^2 in terms of h:
w^2 = λ(676 - h^2)/2kh - h^2
Substituting this into the equation A = hw gives
h^3(sqrt(676 - h^2)) = Ah/k
or
h^4 - (676/k)h + (A/k) = 0
This is a quartic equation that can be solved using a numerical method, such as the Newton-Raphson method. The result is h = 11.19 inches and w = 11.19/sqrt(3) = 6.46 inches (rounded to two decimal places). Therefore, the dimensions of the strongest beam that can be cut from the log are approximately 11.19 inches by 6.46 inches. The answer is (A).

A Wooden Beam Has a Rectangular Cross Section of Height