a hiker is on a cliff with a height of 103 m and creates a $90^{\\circ}$ angle with the ocean. if a boat can…

a hiker is on a cliff with a height of 103 m and creates a $90^{\\circ}$ angle with the ocean. if a boat can see the hiker from an angle of elevation of $36^{\\circ}$, how far is the hiker from the boat?\ndistance from boat = ? m\nround your answer to the nearest hundredth.

a hiker is on a cliff with a height of 103 m and creates a $90^{\\circ}$ angle with the ocean. if a boat can see the hiker from an angle of elevation of $36^{\\circ}$, how far is the hiker from the boat?\ndistance from boat = ? m\nround your answer to the nearest hundredth.

Answer

Explanation:

Step1: Define given values

Cliff height (opposite side to $36^\circ$): $h = 103$ m; angle of elevation $\theta = 36^\circ$; we need the hypotenuse (distance between hiker and boat, $d$).

Step2: Use sine trigonometric ratio

Sine of angle is opposite over hypotenuse: $$\sin(\theta) = \frac{h}{d}$$ Rearrange to solve for $d$: $$d = \frac{h}{\sin(\theta)}$$

Step3: Substitute values and calculate

$$d = \frac{103}{\sin(36^\circ)}$$ Using $\sin(36^\circ) \approx 0.5878$: $$d \approx \frac{103}{0.5878} \approx 175.23$$

Answer:

175.23 m