Question

A vertical tower Stands on a horizontal plane and is surmounted by a vertical flag staff of height h. At a point on the plane, the angles of Elevation of the bottom and the top of the flag staff are `alpha and beta` respectively Prove that the height of the tower is `(htanalpha)/(tanbeta - tanalpha)`

Answer 1

We can draw a diagram with the given details.
Please refer to video for the diagram.
Here, height of the tower ` = BC = x`
`AB = h, CD = y`
`/_CDB = alpha, /_CDA = beta`
`:.tan alpha = (BC)/(CD) `
`=> tan alpha = x/y`
`y = x/ tan alpha->(1)`
Now, `tan beta = (AC)/(CD) `
`=> tan beta = (h+x)/(y)`
`=> y tan beta = h+x`
Putting value of `y` from (1),
`=>x/tan alpha tan beta= h+x`
`=>x(tanbeta/tanalpha-1) = h`
`=>x((tanbeta-tanalpha)/tanalpha) = h`
`=>x = (htanalpha)/(tanbeta-tanalpha)`
So, height of tower is `(htanalpha)/(tanbeta-tanalpha)`.