Question

when seen from to top of a 32 meter high lighthouse, the angle of depression of a boat is 29 degrees. how far away is the boat from the lighthouse?
a). measure of the angle of elevation =

b.) equation/set up:

c.) x =

d ) x≈

Explanation:

Step1: Find angle of elevation

The angle of elevation from the boat to the top of the lighthouse is equal to the angle of depression from the lighthouse to the boat, by alternate interior angles.
$\text{Angle of elevation} = 29^\circ$

Step2: Set up trigonometric equation

Let $x$ be the horizontal distance from the boat to the lighthouse. We use the tangent function: $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$, where the opposite side is the height of the lighthouse ($32$ m) and $\theta = 29^\circ$.
$\tan(29^\circ) = \frac{32}{x}$
Rearranged to solve for $x$: $x = \frac{32}{\tan(29^\circ)}$

Step3: Calculate numerical value

Use $\tan(29^\circ) \approx 0.5543$ to compute $x$.
$x \approx \frac{32}{0.5543}$

Answer:

a) $\boldsymbol{29^\circ}$
b) $\boldsymbol{x = \frac{32}{\tan(29^\circ)}}$
c) $\boldsymbol{x = \frac{32}{\tan(29^\circ)}}$
d) $\boldsymbol{x \approx 57.7}$ meters