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HW 8 Solutions

# HW 8 Solutions - Mathematics 408K Homework 8 Gagan Tara...

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Mathematics 408 K: Homework 8 Gagan Tara Nanda 09 th October, 2004 HW 8 : # s 8 , 10 , 12 , 20 , 21 , 22 8. I’ve clipped the diagram below from the original solutions, so to have it match my question, change the length 5 in the diagram to 8 , since the island in my question is 8 miles from the shore. In this setup, x is the distance of the light beam’s endpoint from the left edge of the shore line. We want to know how fast the beam is travelling along the shore line, so we want to fi nd dx dt . From the diagram, tan θ = x 8 x = 8 tan θ . Di ff erentiate implicitly with respect to t to get dx dt = 8 sec 2 d θ dt . Observe that 1 revolution is 2 π radians, so the searchlight turns through 4 · 2 π = 8 π radians per minute,

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Gagan Tara Nanda 2 which is d θ dt . Hence dx dt = 8 sec 2 θ (8 π ) = 64 π cos 2 θ . At the particular instant requested, the beam makes an angle of 45 with the shoreline, which is 90 θ . So θ = 90 45 = 45 , and we get dx dt = 64 π cos 2 (45 ) = 64 π ³ 1 2 ´ 2 = 128 π miles/min.
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HW 8 Solutions - Mathematics 408K Homework 8 Gagan Tara...

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