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Question
Three villagers A, B and C can see each other using telescope across a valley. The horizontal distance between A and B is 8 km and the horizontal distance between B and C is 12 km. The angle of depression of B from A is 20° and the angle of elevation of C from B is 30°. Calculate the vertical height between A and B. (tan 20° = 0.3640, `sqrt3` = 1.732)
Solution
Let AD is the vertical height between A and B
In the right ∆ABD
tan 20° = `"AD"/"BD"`
0.3640 = `"AD"/8`
AD = 0.3640 × 8 = 2.912 km
∴ AD = 2.91 km
CE is the vertical height between C and B
In the right ∆BCE, tan 30° = `"CE"/"BE"`
`1/sqrt(3) = "CE"/12`
⇒ `sqrt(3)"CE"` = 12
CE = `12/sqrt(3)`
= `(12 xx sqrt(3))/(sqrt(3) xx sqrt(3))`
= `(12 xx sqrt(3))/3`
= `4sqrt(3)`
= 4 × 1.732
= 6.928
= 6.93 km
The vertical height between A and B = 2.91 km
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