A plane flying horizontally at an altitude of 3 miles and a speed of 520 mi/h passes directly over a radar station. Find the rate at which the distance

Question

A plane flying horizontally at an altitude of 3 miles and a speed of 520 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 4 miles away from the station. (Round your answer to the nearest whole number.)



Answer 1

Sketch a right triangle with one leg going vertically a length of 3 miles above the station, and the other leg x going horizontally from the top of the first leg. Distance r from plane to station is related by r² = x² + 3². Differentiate: 2r dr/dt = 2x dx/dt, which rearranges to the desired dr/dt = x/r dx/dt. When r = 4, x² = 4² - 9 and x = √7. So dr/dt = (√7 / 4) * 520 mph = 344 mph.