The length of the shadow of a vertical pole of height `h ,` thrown by the suns rays at three different moments are `h ,2ha n d3h` . Find the sum of the angles of elevation of the rays at these three moments.

Options

A. `pi/2`
B. `pi/3`
C. `pi/4`
D. `pi/6`

Answer

A

Explanation

Let OA be the vertical pole of height h and `OP_1`, `OP_2`, `OP_3` be the lengths of shadow.
In `triangleAOP_1`, we have
`tan theta_1=(OA)/(OP_1)=h/h = 1 rArr theta_1=pi/4`
In `AOP_2`, we have
`tan theta_2 = (OA)/(OP_2)`
`rArr tan theta_2= h/(2h) rArr tan theta_2=1/2 rArr theta_2=tan^(-1)1/2`
Similarly, we have
tan `theta_3=1/3 rArr theta_3=tan^(-1)1/3`
`therefore` Sum of the angles of elevation of the rays =`theta_1+theta_2+theta_3`

`=pi/4+tan^(-1) 1/2 + tan^(-1)1/3=pi/2 + tan^(-1)((1/2+1/3)/(1-1/2xx1/3))`
`=pi/4+tan^(-1) 1= pi/4+pi/4 = pi/2`