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Expert-verified Found in: Page 215 ### Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974 # Leg traction. The accompanying figure shows how a leg may be stretched by a pulley line for therapeutic purposes. We denote by ${{\mathbit{F}}}_{{\mathbf{1}}}$ the vertical force of the weight. The string of the pulley line has the same tension everywhere. Hence, the forces role="math" localid="1659529616162" ${{\mathbit{F}}}_{2}$ and ${{\mathbit{F}}}_{{\mathbf{3}}}$ have the same magnitude as ${{\mathbit{F}}}_{{\mathbf{1}}}$. Assume that the magnitude of each force is 10 pounds. Find the angle ${\mathbit{\theta }}$ so that the magnitude of the force exerted on the leg is 16 pounds. Round your answer to the nearest degree. (Adapted from E. Batschelet, Introduction to Mathematics for Life Scientists, Springer, 1979.) The angle $\theta =2arcoos\left(0.8\right)\approx 74°$

See the step by step solution

## Step 1: Draw the diagram below

The diagram is shown below which represents the given condition. ## Step 2: Find the angle

Now, from the above diagram:

$||{F}_{2}+{F}_{3}||=2\mathrm{cos}\left(\frac{\theta }{2}\right)||{F}_{2}||=2\mathrm{cos}\left(\frac{\theta }{2}\right)$

It is required that:

$||{F}_{2}+{F}_{3}||=16\phantom{\rule{0ex}{0ex}}2\mathrm{cos}\left(\frac{\theta }{2}\right)=16$

$\theta$role="math" localid="1659529396473" $=2arcoos\left(0.8\right)\approx 74°$

Hence, the required angle is $\theta$$=2arcoos\left(0.8\right)\approx 74°$ ### Want to see more solutions like these? 