A sailboat that is 2 km due west of a lighthouse sends a signal to the lighthouse that it is in distress. The lighthouse quickly signals a rescue plane that is 7 km due south of the lighthouse. What heading from due north should the plane take in order to intercept the troubled sailboat? Round your answer to the nearest degree.
[tex]
\begin{array}{c}
\Rightarrow a=2 km \\
b=7 km \\
\angle \theta=? \\
\tan \theta=\frac{2}{7} \\
\theta=\tan ^{-1}\left(\frac{2}{7}\right)
\end{array}
[/tex]
Answer (2)
Define the problem: A plane needs to intercept a sailboat, and we need to find the heading from due north.
Calculate the angle α between the south direction and the line connecting the plane to the sailboat using tan ( α ) = 7 2 , which gives α ≈ 15.94 5 ∘ .
Determine the heading from due north by adding 9 0 ∘ to α : 9 0 ∘ + 15.94 5 ∘ = 105.94 5 ∘ .
Round the result to the nearest degree: 10 6 ∘ .
Explanation
Problem Analysis and Setup The sailboat is 2 km west of the lighthouse, and the rescue plane is 7 km south of the lighthouse. We need to find the heading from due north that the plane should take to intercept the sailboat. Let's visualize this as a right triangle where the distance west is one leg, the distance south is the other leg, and the direct path from the plane to the sailboat is the hypotenuse. The angle we need to find is the angle between the north direction and the hypotenuse.
Finding the Angle Alpha Let's denote the angle between the south direction and the line connecting the plane to the sailboat as α . We can use the tangent function to find this angle: tan ( α ) = adjacent opposite = distance south distance west = 7 2
Calculating Alpha Now, we find the angle α by taking the inverse tangent (arctan) of 7 2 .
α = arctan ( 7 2 ) Using a calculator, we find that: α ≈ 15.94 5 ∘
Finding the Heading The heading from due north is the angle 9 0 ∘ + α , since the plane is south of the lighthouse and needs to head west to intercept the sailboat. Heading = 9 0 ∘ + α = 9 0 ∘ + 15.94 5 ∘ = 105.94 5 ∘ Rounding to the nearest degree, the heading is approximately 10 6 ∘ .
Final Answer Therefore, the rescue plane should take a heading of approximately 10 6 ∘ from due north to intercept the sailboat.
Examples
Pilots use headings to navigate their aircraft. If an air traffic controller needs a pilot to intercept another aircraft, they will provide the pilot with a heading to fly. This ensures the pilot flies in the correct direction to reach the other aircraft efficiently. In this case, the calculation we performed helps determine the precise heading needed for the rescue plane to reach the sailboat.
To intercept the sailboat, the rescue plane should head approximately 106 degrees from due north. This is calculated by using tangent to find the angle between the line to the sailboat and the south direction. After determining this angle, we add it to 90 degrees to find the necessary heading from due north.
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