Multiple choice

The angle of depression of a point on the ground from the top of a tree is $60^{\circ}$. Lowering down 20 m from the top. the angle of depression changes to $30^{\circ}$. Find the distance between the point and the foot of the tree. Also find the height of the tree.

  1. 10 $\sqrt{3}$ ,20
  2. 20 $\sqrt{3}$ ,30
  3. 10 $\sqrt{3}$ ,30
  4. 20 $\sqrt{3}$ ,40
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C Correct answer
Explanation

Let h be the height of the tree and d be the distance. tan 60 = h/d => h = d*sqrt(3). tan 30 = (h-20)/d => d*tan 30 = h-20. Substituting h: d/sqrt(3) = d*sqrt(3) - 20 => 20 = d(sqrt(3) - 1/sqrt(3)) = d(2/sqrt(3)) => d = 10*sqrt(3). h = 10*sqrt(3)*sqrt(3) = 30.

AI explanation

Let the total height of the tree be h and the horizontal distance to the point be d. From the lower point 20 m above the ground, tan(30) equals d divided by (h minus 20), meaning d equals (h minus 20) multiplied by the square root of 3. From the top of the tree, tan(60) equals d divided by h, meaning d equals h multiplied by the square root of 3. Setting the two equations for d equal gives h multiplied by the square root of 3 equals (h minus 20) multiplied by the square root of 3, which solves to h equals 30 m and d equals 10 times the square root of 3.

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