math

ou are a pirate captain trying to determine how long it will take you to get to an island where you expect to find buried treasure. You are heading due North, toward the island. 60° West of North, you spot a light house. According to your sea chart, the top of the lighthouse is 50 m off the ground. You can see the top of the light house at a 5° incline to the horizon. The sea chart also tells you that the distance from the lighthouse to the island is 512.39 m. If the boat is travelling at 5 m/s, how long will it take for the boat to reach the island?

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  1. Draw a diagram. Let
    T be the top of the lighthouse
    B be the bottom of the lighthouse
    S be the location of the ship
    I be the location of the island

    Then SB, the distance to the lighthouse can be found using
    50/SB = tan5°
    SB = 571.50 m

    Now, angle θ=BIS can be found using
    sinθ/571.50 = sin60°/512.39
    sinθ = 0.9659
    θ = 75°
    That means that angle SBI = 45°

    So, the distance SI to the island is
    SI^2 = 572.39^2 + 571.50^2 - 2*572.39*571.50*cos45° = 191624
    SI = 437.75 m

    So, at 5m/s, it will take 437.75/5 = 87.55 seconds to arrive at the island

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