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3 years ago

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A rectangular beam is cut from a cylindrical log of radius 25 cm. A cross-section of the log is shown below. The strength, S, of

a beam of width w and height h is proportional to wh2.
(a) Express the strength of the beam as a function of w only. Use k as the proportionality constant.
S(w) =

(b) Find the exact width and height of the beam of maximum strength.
Width =
Height =

1 answer:

Delicious77 [7]3 years ago

8 0

We are given the statement that the strength, S, of a beam of width w andheight h is proportional to wh2. hence using k as the proportionality constant, the equation is S = k wh2. the exact width and height can be determined by applying optimization to the problem. optimization is taking the first derivative of an equation and equating then to zero

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$\begin{aligned} A^{\prime}(t) &= 3\, t^{2} - 15\, t + 12 \\ &= 3\, (t - 1) \, (t + 4)\end{aligned}$ .

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Therefore $\! t = 1$ would be the only local maximum over the interval $0 \le t \le 6\!$ .

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