Multiple choice

A man on the deck of a ship is $12m$ above water level. he observes that the angle of elevation, of the top of a cliff is ${45}^{o}$ and the angle of depression of its base is ${30}^{o}$. Calculate the distance of the cliff from the ship and the height of the cliff.

  1. Height$=36m$; Distance$=29m$
  2. Height$=46.256m$; Distance$13.759m$
  3. Height$=40m$; Distance$=25m$
  4. Height$=32.784m$; Distance$=20.784m$
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D Correct answer
Explanation

Let distance be d. Height of cliff = h. From ship (12m above water), angle of elevation to top is 45, so h-12 = d * tan(45) = d. Angle of depression to base is 30, so 12 = d * tan(30) = d * (1/sqrt(3)). d = 12 * sqrt(3) = 20.784m. h = 12 + 20.784 = 32.784m.

AI explanation

Let the horizontal distance of the cliff from the ship be d. For the angle of depression of 30 degrees to the waterline, tan(30 degrees) equals 12 divided by d, so d equals 12 times the square root of 3, which is approximately 20.784 meters. The height of the cliff above water level is the ship's deck height plus the elevation to the top, calculated as 12 plus d times tan(45 degrees). Since tan(45 degrees) is 1, the total cliff height is 12 plus 20.784, yielding 32.784 meters.

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