how fast is the balloon rising actually in the following scenario?
A balloon is rising straight up. My head is rising at a rate of 16 degrees per second to look at it. If I am standing 25 m from a point below the balloon?
The line joining the balloon to the ground and the line joining the point below the balloon that lies at the height of the person's head to the person's head are perpendicular to each other. If the height of the balloon from the level of the person's head is h, the tan of the angle that the line joining the person's head and the balloon makes with the horizontal line is `tan^-1(h/25)`
`x = tan^-1(h/25)`
`tan x = h/25`
Use implicit differentiation with respect to t
`sec^2x*(dx/dt) = (1/25)*(dh/dt)`
The value of `(dx)/(dt) = 16` degrees/second
=> `(dh)/(dt) = 25*sec^2 x*16 = 32*sec^2x`
The balloon is rising up at `32*sec^2x` m/s where x is the angle of the head at that moment of time.