calculus

A rectangular beam is cut from a cylindrical log of radius 25 cm. The strength of a beam of width w and height h is proportional to wh2. (See Figure 4.70.) Find the width and height of the beam of maximum strength.

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1. Draw a diagram. Given an inscribed rectangle of width w and height h, the diagonal of the rectangle is a diameter of the circle.

w^2 + h^2 = d^2
h^2 = 2500 - w^2

s = kwh^2 = kw(2500-w^2) = 2500kw - kw^3
s' = 2500k - 3kw^2

max strength is where s' = 0

3w^2 = 2500
w = 28.87

Now you can find h.

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2. A rectangular beam is cut from a cylindrical log of radius r=10 cm. The strength S of a beam of width w and height h is given by
S=wh2
The cross section of the log and beam are shown in the figure below.

Determine the width and height of the beam of maximum strength that can be cut from the log. State your answers rounded to 2 decimal places.

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