A ship is sailing directly toward a lighthouse at a constant speed of 15 knots. The angle of elevation to the top of the lighthouse is 30 degrees. Determine the rate at which the distance between the ship and the lighthouse is decreasing when the ship is 3 nautical miles away.
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Let's denote the distance between the ship and the lighthouse as D, and the rate at which the distance is decreasing as dD/dt.
We can use the concept of trigonometry to relate the angle of elevation, the distance, and the height of the lighthouse.
In a right triangle formed by the ship, the lighthouse, and a line perpendicular to the ground, the angle of elevation to the top of the lighthouse is 30 degrees. The distance D is the hypotenuse, and the height of the lighthouse is the opposite side.
Using the definition of sine, we have:
sin(30 degrees) = height / D
Since sine(30 degrees) = 1/2, we can rewrite the equation as:
1/2 = height / D
Multiplying both sides by D, we get:
D/2 = height
We also know that the height of the lighthouse is constant, so the rate at which the height changes is 0 (dh/dt = 0).
Differentiating the equation D/2 = height with respect to time, we get:
(1/2) * dD/dt = dh/dt
Since dh/dt = 0, we can solve for dD/dt:
(1/2) * dD/dt = 0
dD/dt = 0
Therefore, the rate at which the distance between the ship and the lighthouse is decreasing when the ship is 3 nautical miles away is 0 knots.
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