Answer:
173.6 ft
Step-by-step explanation:
The given scenario can be modelled as two right triangles, where the height of both triangles represents the combined height of the lighthouse and cliff (195 ft).
The base of the cliff is point D, and boat A is closer to the cliff than boat B. Therefore, the base of the first triangle is DA, and the base of the second triangle is DA + AB.
The angle of depression from the top of the lighthouse to boat A is 57°, and to boat B is 33°.
To find the distance between the boats (AB), 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]
Triangle ADC: Boat A
Given values:
Substitute the values into the tangent ratio and solve for the exact value of DA:
[tex]\tan 57^{\circ}=\dfrac{195}{DA}\\\\\\DA=\dfrac{195}{\tan 57^{\circ}}[/tex]
Triangle BDC: Boat B
Given values:
- θ = 33°
- O = 195
- A = DA + AB
Substitute the values into the tangent ratio and solve for AB:
[tex]\tan 33^{\circ}=\dfrac{195}{DA+AB}\\\\\\DA+AB=\dfrac{195}{\tan 33^{\circ}}\\\\\\AB=\dfrac{195}{\tan 33^{\circ}}-DA[/tex]
Now, substitute the expression for DA into the equation for AB:
[tex]AB=\dfrac{195}{\tan 33^{\circ}}-\dfrac{195}{\tan 57^{\circ}}\\\\\\AB=173.639187270...\\\\\\AB=173.6\; \sf ft\;(nearest\;tenth)[/tex]
Therefore, the distance between the two sailboats is 173.6 ft.
