# A hawk flying at 19 m/s at an altitude of 165 m accidentally drops its prey. How far does the prey travel if its trajectory is a parabola?   The parabolic trajectory of the falling prey is described by the equation `y = 165 - 165/57x^2` until it hits the ground, where y is its height above the ground and x is the horizontal distance traveled in meters. Calculate the distance traveled by the prey from the time it is dropped until the time it hits the ground.

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The distance travelled is the arclength of the parabola `y = 165-165(x^2/57)` between `x=0` and the place where the prey hits the ground.

The prey hits the ground when `y=0`

ie when `x =sqrt(57)`

Write the parabola in terms of the parameter t:

`x = 2at + k`

`y = at^2 + h`

Since ` ``(y-h) = (x-k)^2/(4a)`

`implies a = -57/(4(165)) = -0.086`  `k= 0` and `h =165`

The arclength of a parabola is given by

`s = a(tsqrt(1+t^2) +sinh^(-1)t)`

When ` `` ``x = sqrt(57)` , `t = sqrt(57)/(2a) = -sqrt(57)/(2*0.086) = -43.71`

`implies s = 165.43 m`

The prey travels 165.43 metres parabolically

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