The lower 24 m portion of a 50 m tall tower is painted green and the remaining portion red. What is the distance of a point on the ground from the base of the tower where the two different portions of the tower subtend equal angles?

Options

A. 60m
B. 72m
C. 90m
D. 120m

Answer

D

Explanation

Let the distance , be x, and angle
`APB=theta`, then `angleBPC=angleAPC=theta//2`
In triangle `Delta APB`,
`tan theta = (AB)/(x)=(50)/(x) " " ...(1)`
and in triangle APC
`tan (theta)/(2)=(AC)/(x)=(24)/(x)` ...(2)
`tan theta = (2 tan (theta)/(2))/(1-tan^(2) (theta)/(2)) ` ...(3)
Putting the value of `tan theta` and `tan (theta)/(2)`
From equation (1) and(2) in equation (3),
`(50)/(x)=(2xx(24)/(x))/(1-((24)/(x))^(2))`
or, `(50)/(x)=(48)/(x) xx(x^(2))/(x^(2)-(24)^(2))`
or, `50 {x^(2)-(24)^(2)}=48x^(2)`
or, `50x^(2)-50xx(24)^(2)=48x^(2)`
or, `2x^(2)=(24)^(2)xx50`
`x=25xx(24)^(2)`
`x=|5xx24|=120m`
Such a point can exist on the either side of the tower.