Multiple choice

From a light house the angle of depression of two ships on opposite sides of the light house are observed to be 30 and 45. If the height of the light house be 100 metres, then the distance between the ships of the line joining them passes through the foot of the light house is 100 $\sqrt{3 - 1}$.

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The horizontal distances from the foot of the lighthouse are 100 cot 30° = 100sqrt(3) metres and 100 cot 45° = 100 metres. Since the ships are on opposite sides, their separation is 100sqrt(3) + 100 = 100(sqrt(3) + 1) metres, not 100sqrt(3 - 1).

AI explanation

Let the height of the lighthouse be 100 meters, and let the ships be at distances x and y from its foot. Using the tangent of the angles of depression, tan 45 degrees equals 100 divided by x, meaning x is 100 meters; similarly, tan 30 degrees equals 100 divided by y, meaning y is 100 times the square root of 3. The total distance between the ships is x plus y, which equals 100 plus 100 times the square root of 3, or 100 multiplied by the sum of 1 and the square root of 3. The given expression of 100 multiplied by the difference of 3 and 1 is incorrect, so the statement is false.

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