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a151t2au - M ATH 151 Answers to Test 2 Autumn 2009 1 A...

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MATH 151 Answers to Test 2 Autumn 2009 1. A baseball diamond is a square with sides 90 feet long. A baseball player ( P in the figure below) is advancing from the second base ( S ) to the third base ( T ) and the umpire at the home plate ( H ) is watching him. Let θ be the angle between the third baseline ( TH ) and the line of sight from the umpire to the player ( HP ). How fast is θ changing when the runner is 30 feet from the third base and running at 24 feet per second? Let x be the distance PT from the player to the third base in feet. In the triangle HTP , TH = 90 ft is constant. The angle PTH = θ and x are changing. tan θ = x/ 90; so 90 tan θ = x . Di ff erentiate both sides with respect to time we get 90 sec 2 θ d θ dt = dx dt . When x = 30, tan θ = 30 / 90 = 1 / 3, and so sec 2 θ = 1 + tan 2 θ = 1 + (1 / 3) 2 = 10 / 9. As dx dt = 24, 90 × 10 9 d θ dt = 24. Hence d θ dt = 24 100 . Answer : 0 . 24 radians per second H F S T P x 90 θ 2. Find the equation of the tangent line to the graph of the equation at the specified point.
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