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Math Help - Differentiate

  1. #1
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    Differentiate

    A man in a boat is 3 km from O, which is the nearest point in straight beach. His destinations is 6 km along the beach from O. If he can row at 4km/h and walk at 5km/h towards what point on the beach should he row to reach his destination in the least possible time? Ans: 4km from O
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  2. #2
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    Quote Originally Posted by tommylai12 View Post
    A man in a boat is 3 km from O, which is the nearest point in straight beach. His destinations is 6 km along the beach from O. If he can row at 4km/h and walk at 5km/h towards what point on the beach should he row to reach his destination in the least possible time? Ans: 4km from O
    Code:
     
    o B
    |\
    | \
    |  \
    |   \
    |    \
    o-----o---------o beach
    O     X         D
    B = boat
    D = destination
    OB = 3 km
    OD = 8 km
    Let OX = x km.
    Time to reach X from B: t_1(x)=\frac{\sqrt{BO^2 + OX^2}}{4}=\frac{\sqrt{9 + x^2}}{4}.
    Time to reach D from X: t_2(x)=\frac{XD}{5}=\frac{8-x}{5}
    Total time: t(x)=t_1(x)+t_2(x)=\frac{\sqrt{9 + x^2}}{4}+\frac{8-x}{5}, where x varies from 0 to 8.

    Differentiate t(x) w.r.t. x, then solve t'(x)=0. The answer should be x=4.
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