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Calculus

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A car leaves an intersection traveling west. Its position 5 sec later is 22 ft from the intersection. At the same time, another car leaves the same intersection heading north so that its position 5 sec later is 27 ft from the intersection. If the speed of the cars at that instant of time is 15 ft/sec and 5 ft/sec, respectively, find the rate at which the distance between the two cars is changing. (Round your answer to one decimal place.)
ft/sec

  • Calculus - ,

    let the distance of the west-bound car from the intersection be x
    let the distance of the north-bound car from the intersection be y
    and let the distance between them be d

    d^2 = x^2 + y^2
    2d dd/dt = 2x dx/dt + 2y dy/dt
    d dd/dt = x dx/dt + y dy/dt

    when x=22, y=27 , dx/dt = 15 ft/sec , dy/dt = 5 ft/sec
    and d^2 = 22^2 + 27^2 = 1213
    d = √1213

    √1213 dd/dt = 22(15) + 27(5) = 465

    dd/dt = 465/√1213 = appr 13.4 ft/s

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