Q41 of 68 Page 9

There is a small island in the middle of 100 m wide river. There is a tall tree on the island. Points P and Q are points directly opposite each other on the two banks and in line with the tree. If the angles of elevation of the top of the tree at P and Q are 30° and 45°, find the height of the tree.


In right Δ ABP, we have




⇒ x = √3h …(i)


In the right Δ ABQ, we have




⇒ 100 – x = h


⇒ 100 = h + x


⇒ 100 = h + √3h [from (i)]


⇒ 100 = h(√3 + 1)



Multiplying and divide by the conjugate of √3 + 1, we get



[∵ (a – b)(a + b) = (a2 – b2)]



⇒ h = 50 (√3 – 1)


⇒ h = 50 (1.732 – 1)


⇒ h = 50 (0.732)


⇒ h = 36.6m


Hence, the height of the tree is 36.6m


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