Car A is traveling north on a straight highway and car B is traveling west on a different straight highway. Each car is approaching the inte

Question

Car A is traveling north on a straight highway and car B is traveling west on a different straight highway. Each car is approaching the intersection of these highways. At a certain moment, car A is 0.3 km from the intersection and traveling at 95 km/h while car B is 0.4 km from the intersection and traveling at 90 km/h. How fast is the distance between the cars changing at that moment? km/h

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Linh Đan 6 months 2021-08-09T11:21:40+00:00 2 Answers 15 views 0

Answers ( )

    0
    2021-08-09T11:23:03+00:00

    Answer:

    129 km/hr

    Explanation:

    Distance of Car A North of the Intersection, y=0.3km

    Distance of Car B West of the Intersection, x=0.4 km

    The distance z,  between A and B is determined by the Pythagoras theorem

    z^2=x^2+y^2

    z^2=0.4^2+0.3^2=0.25\\z=\sqrt{0.25}=0.5km

    Taking derivative of z^2=x^2+y^2

    2z\frac{dz}{dt}= 2x\frac{dx}{dt}+2y\frac{dy}{dt}

    \frac{dx}{dt}=90km/hr, \frac{dy}{dt}=95km/hr

    2(0.5)\frac{dz}{dt}= 2(0.4)X90+2X0.3X95\\\frac{dz}{dt}=72+57=129

    The distance z, between the cars is changing at a rate of 129 km/hr.

    0
    2021-08-09T11:23:13+00:00

    Answer:

    The rate at which the distance changes at that moment is 91.743  Km/h

    Explanation:

    The time for car a to get to the intersection = \frac{Distance}{Speed}

                     = \frac{0.3}{95}

                    = 0.00316 h

    The time for car B to get to the intersection = \frac{Distance}{Speed}

                    = \frac{0.4}{90}

                    = 0.00444 h

    Total time = \sqrt{0.00316^{2} + 0.00444^{2} }

                    = 0.00545 h

    Total distance between the two cars at that moment = \sqrt{0.3^{2} + 0.4^{2}  }

                    = 0.5 Km

    The rate at which the distance between the cars is changing = \frac{0.5}{0.00545}

                    =  91.743  Km/h

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