The top of the Fastnet lighthouse, F, is 49 m above sea level - Leaving Cert Mathematics - Question d - 2022

To determine the horizontal distance, we will use the tangent function in trigonometry. The tangent of an angle in a right triangle is defined as the ratio of the opposite side to the adjacent side.

1. Identify the relevant information from the problem:

   • Height of the lighthouse (opposite side) = 49 m
   • Angle of elevation = 1°2'

2. Write the equation that relates the tangent of the angle to the opposite and adjacent sides: $an(1^{ ext{o}} 2') = \frac{49}{d}$

3. Rearrange this equation to solve for d: $d = \frac{49}{\tan(1^{ ext{o}} 2')}$

4. Calculate the value of $an(1^{ ext{o}} 2')$ using a calculator:

   • Convert the angle 1°2' to decimal form:
     • $1^{ ext{o}} 2' = 1 + \frac{2}{60} = 1.0333^{ ext{o}}$
   • Now compute $an(1.0333^{ ext{o}}) eq 0.0175$

5. Plug this value back into the equation for d:

   $$$$

eq 2799 ext{ m}$$

6. Convert distance from meters to kilometers: $d = \frac{2799}{1000} = 2.799 ext{ km}$

7. Round to 2 decimal places: $d \approx 2.80 ext{ km}$