Answer:
281.5 ft
Step-by-step explanation:
The given scenario can be modelled as a right triangle where:
- The height of the triangle represents the height of the lighthouse (65 ft).
- The base of the triangle represents the distance between the windsurfer and the base of the lighthouse.
- The angle between the base of the triangle and its hypotenuse represents the angle of elevation from the windsurfer to the top of the lighthouse (13°).
To find the distance (d) between he windsurfer and the base of the lighthouse, we can use the tangent trigonometric ratio:
[tex]\boxed{\begin{array}{l}\underline{\textsf{Tangent trigonometric ratio}}\\\\\sf \tan(\theta)=\dfrac{O}{A}\\\\\textsf{where:}\\\phantom{ww}\bullet\;\textsf{$\theta$ is the angle.}\\\phantom{ww}\bullet\;\textsf{$O$ is the side opposite the angle.}\\\phantom{ww}\bullet\;\textsf{$A$ is the side adjacent the angle.}\end{array}}[/tex]
In this case:
Substitute the values into the tangent ratio and solve for d:
[tex]\tan 13^{\circ}=\dfrac{65}{d} \\\\\\ d=\dfrac{65}{\tan 13^{\circ}} \\\\\\ d=281.5459318284...\\\\\\ d=281.5\; \sf ft\;(nearest\;tenth)[/tex]
Therefore, the windsurfer is 281.5 ft from the shore.
