Related Rates Problems
Date: 01/15/99 at 23:11:48 From: Hiu Sze Subject: Related Rate 1) A plane flying horizontally at an altitude of 1 mi and a speed of 500 mi/h passes directly over a radar station. Find the rate at which the distance from the plane to the station is increasing when it is 2 miles away from the station. Answer: The plane's altitude is 1 mile and its velocity is 500 mi/h. 2) At noon, ship A is 150km west of ship B. Ship A is sailing east at 35 km/h and ship B is sailing north at 25 km/h. How fast is the distance between the ships changing at 4:00 p.m.? Thank you! Hiu Sze
Date: 01/16/99 at 10:31:31 From: Doctor Anthony Subject: Re: Related Rate Question 1: Let x = the horizontal distance that the plane travels from a point directly overhead. Let y = the slant distance from the radar station to the plane. Then by the Pythagorean Theorem, y^2 = x^2 + 1^2 and differentiating with respect to t: 2y(dy/dt) = 2x(dx/dt) and dx/dt = 500 dy/dt = (x/y)(500) When x = 2, y = sqrt(4+1) = sqrt(5), so x/y = 2/sqrt(5) and dy/dt = 1000/sqrt(5) = 447.21 mph So the distance between the plane and the radar station is increasing at 447.21 mph. Question 2: Let x = the distance of ship A west of the north-south line through B. y = the distance of ship B north of the east-west line through A. At any time the distance apart is given by L^2 = x^2 + y^2. Then: 2L dL/dt = 2x dx/dt + 2y dy/dt L dL/dt = x(-35) + y(25) dL/dt = [-35x + 25y]/L At 4:00 p.m.: x = 150-140 = 10 y = 100 L^2 = 100+10000 = 10100 => L = 100.4988 So: dL/dt = [-350 + 2500]/100.4988 = 21.393 km/hr So at 4:00 p.m., the ships are separating at a speed of 21.393 km/hr. - Doctor Anthony, The Math Forum http://mathforum.org/dr.math/
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