A hawk flying at 19 m/s at an altitude of 165 m accidentally drops its prey. The parabolic trajectory of the falling prey is described by the equation `y = 165- x^2/57` 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 equation is

`d = 165-1/57x^2`

This is a parabola with focal length `f=|-57/4| = 57/4`

We want to calculate the arclength from `x=0` to  `x=sqrt((57)(165)) = 96.98`

Let `h = p/2`  where `p` is the distance along the x-axis from the vertex to the point where we are...

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The distance equation is

`d = 165-1/57x^2`

This is a parabola with focal length `f=|-57/4| = 57/4`

We want to calculate the arclength from `x=0` to  `x=sqrt((57)(165)) = 96.98`

Let `h = p/2`  where `p` is the distance along the x-axis from the vertex to the point where we are measuring the arclength

`implies` `h = 96.98/2 = 48.49 `

` ` and let `q = sqrt(f^2+h^2) = 50.54`

Then the arclength `s` satisfies

`s = ((hq)/f) + fln((h+q)/f) = 199.6m` (look for length of an arc of a parabola in reference below)

Alternatively, evaluate `s = int_0^96.98 sqrt(1+((dy)/(dx))^2) dx` (this involves integrating `sec^3u` after making the substitution `4/57x = tanu`)

The prey travels 199.6m

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