The man climbing up an incline moves horizontally at 4 m/s. He moves up the incline at 8 m/s.
Take the angle that the incline makes with the horizontal as `theta` . As the man climbs up the incline at 8 m/s, he moves horizontally at 4 m/s. In one second, the distance moved up the slope is 8 m and the distance moved in the horizontal direction is 4 m.
This gives `cos theta = 4/8 = 1/2`
=> `theta = cos^-1(1/2)`
=> `theta` = 60 degrees
`sin 60 = sqrt 3/2` = (distance moved vertically upwards)/(distance climbed up the incline)
The rate at which the height of the person increases is equal to `(8*sqrt 3)/2 = 4*sqrt 3` m/s
The rate at which the person's height increases as he rises up the slope is `4*sqrt 3` m/s