Question

From the top of a lighthouse \[210\] feet high, the angle of depression of a boat is \[27\] degrees. How do you find the distance from the boat to the foot of the lighthouse where the lighthouse was built at sea level?

Answer 1

Hint: To solve this question, first we will create a correct diagram taking the boat as a point and the lighthouse as a perpendicular line. Then convert the word problem into the mathematical equations. After that using the triangle properties of trigonometric ratios, we will find the distance from the boat to the foot of the lighthouse.
Property used:
\[\tan \theta = \dfrac{{perpendicular}}{{base}}\]

Complete step-by-step answer:
First, let’s create a correct diagram taking boat as a point and the lighthouse as a perpendicular line

Let A be the boat
BC be the light house which is equals to \[210{\text{ feet}}\]
And \[\angle XCA\] is the angle of depression of a boat which is equals to \[27^\circ \]
Now, we have to find the distance from the boat to the foot of the lighthouse which means we have to find the distance AB.
If we see a triangle formed by the lighthouse, the ground, and the boat.
And we know that the angle of depression is equal to the angle of elevation at the boat.
So, by using alternate interior angle property, we can say that
\[\angle XCA = \angle CAB = 27^\circ \]
Now, we know that the sum of interior angle of a triangle is \[180^\circ \]
Therefore, \[\angle ABC + \angle BCA + \angle CAB = 180^\circ \]
On substituting the values, we get
\[90^\circ + \angle BCA + 27^\circ = 180^\circ \]
\[\angle BCA = 63^\circ \]

Now, the distance of the boat is the side perpendicular to the angle of \[63^\circ \] while the height of the lighthouse is the base side.
And we know that
\[\tan \theta = \dfrac{{perpendicular}}{{base}}\]
Therefore, we get
\[\tan 63^\circ = \dfrac{{AB}}{{210}}\]
\[ \Rightarrow AB = \tan 63^\circ \times 210\]
Since \[\tan 63^\circ = 1.963\]
Therefore, we get
\[ \Rightarrow AB = 1.963 \times 210\]
\[ \Rightarrow AB = 412.23{\text{ feet}}\]
Hence, the distance from the boat to the foot of the lighthouse is \[412.23{\text{ feet}}\]
So, the correct answer is “ \[412.23{\text{ feet}}\]”.

Note: The term angle of depression means the angle from the horizontal downward to an object. To speed up the calculation in future, always remember the fact that the angle of depression is always measured from the horizontal and is not in the triangle. Also note that the angle of elevation and the angle of depression are always equal.