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.