From a point on a lighthouse 76 metres above sea level,the angle of depression of a fishing boat 28 degrees.How far is the boat from the foot of the lighthouse?
right-angled triangle trig .
tan 28° = 76/x
x = 76/tan28 = appr 142.9 m
To find the distance between the fishing boat and the foot of the lighthouse, we can use trigonometry.
First, let's draw a diagram to visualize the situation:
Lighthouse (76m)
\ |
\ |
\ |
\ |
\ |
\ |
______\______|______
B |
|| |
|| | Sea Level
_______|______
In this diagram, the point B represents the fishing boat, and the angle \( \theta \) represents the angle of depression.
From the diagram, we see that we have a right triangle with the hypotenuse being the distance from the fishing boat to the foot of the lighthouse.
Now, let's use the trigonometric function tangent (\( \tan \)) to find the distance. The tangent of an angle is defined as the ratio of the length of the opposite side to the length of the adjacent side.
In this case, the angle of depression is given as 28 degrees, and the opposite side is the height of the lighthouse (76m). We can set up the equation as follows:
\( \tan(28^\circ) = \frac{76}{x} \)
where x represents the distance from the foot of the lighthouse to the fishing boat.
To find the value of x, we can rearrange the equation:
\( x = \frac{76}{\tan(28^\circ)} \)
Now, let's calculate the value of x:
\( x = \frac{76}{\tan(28^\circ)} \)
\( x = \frac{76}{\tan(28^\circ)} \approx 149.34 \) meters (rounded to two decimal places)
Therefore, the fishing boat is approximately 149.34 meters away from the foot of the lighthouse.