Question

From the top of a cliff, the angle of depression of a boat on the sea is 60°. If the top of the cliff is 25m above the sea level, calculate the horizontal distance from the bottom of the cliff to the boat.​

Answer 1

Let's denote the horizontal distance from the bottom of the cliff to the boat as x. We can then use trigonometry to solve for x.

In a right triangle where one angle is 60°, the opposite side to the angle of 60° is half the length of the hypotenuse. Therefore, if we let h be the distance from the boat to the bottom of the cliff, we have:

tan(60°) = h / x

tan(60°) is equal to the square root of 3, so we can simplify the equation to:

sqrt(3) = h / x

We also know that the height of the cliff is 25 meters. Therefore, we can write:

h = x + 25

Substituting h in terms of x, we get:

sqrt(3) = (x + 25) / x

Multiplying both sides by x, we get:

sqrt(3) x = x + 25

Subtracting x from both sides, we get:

sqrt(3) x - x = 25

Factoring out x, we get:

x (sqrt(3) - 1) = 25

Dividing both sides by (sqrt(3) - 1), we get:

x ≈ 25 / (sqrt(3) - 1)

Simplifying the denominator by multiplying both the numerator and denominator by (sqrt(3) + 1), we get:

x ≈ 25 (sqrt(3) + 1) / ((sqrt(3) - 1) (sqrt(3) + 1))

x ≈ 25 (sqrt(3) + 1) / 2

x ≈ 21.65 meters (rounded to two decimal places)

Therefore, the horizontal distance from the bottom of the cliff to the boat is approximately 21.65 meters.