Multiple choice

The angle of elevation $\theta$ of the top of a light-house as seen by a person on the ground is such that $\tan{ \theta}=\dfrac{5}{12}$. When the person moves a distance $240\ m$ towards the light-house, the angle of elevation become $\phi$ such that $\tan{\phi}=\dfrac{3}{4}$, Find the height of the light house.

  1. $225\ m$
  2. $265\ m$
  3. $286\ m$
  4. $298\ m$
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A Correct answer
Explanation

Let h be height, d be distance from lighthouse. tan(theta) = h/d = 5/12 => d = 12h/5. tan(phi) = h/(d-240) = 3/4 => d-240 = 4h/3. Substitute d: 12h/5 - 4h/3 = 240. (36h - 20h)/15 = 240. 16h = 3600. h = 225.

AI explanation

Let the initial distance from the person to the lighthouse base be x and its height be h. From the given tangent values, we have h divided by x equals 5 divided by 12, meaning x equals 12h divided by 5. After moving 240 meters closer, h divided by (x minus 240) equals 3 divided by 4, and substituting our expression for x gives h divided by ((12h divided by 5) minus 240) equals 3 divided by 4. Solving this linear equation for h yields 3(12h minus 1200) equals 48h, which simplifies to 36h minus 3600 equals 48h, giving a height of 225 meters.

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