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Find the condition for the borderline case

A fountain has jets that squirt water in all directions at a speed of 16 ft/sec from a central point as shown. The dotted curve shows the boundary between the "dry" region (which the fountain doesn't reach) and the "wet" region reached by the water spray. Find the shape of this dotted curve, giving an equation for it (y as a function of x). Ignore any air resistance, and assume that g is exactly 32 ft/sec2.

Hint: Given the starting angle θ of a drop of water, write down y and x as a function of t, and eliminate t to get y as a function of x. Use some trigonometric identities to rewrite this equation so that it only involves tanθ and not the sine or cosine. Now suppose that you are at some point (x,y) in space. Show how to find the starting angle θ such that the water will reach that point. You will have a quadratic equation for tanθ which sometimes has real solutions (meaning the water will get to you) and sometimes has no real solutions (meaning the water won't get to you). Find the condition for the borderline case.

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The explanations are in the attached PDF file.

For possible reference by Brainmass, I also past in the TEX script from which the pdf was produced - you do not have to read it as you already have the pdf file.

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Here is the plain TEX source

centerline{bf Fountain}

We assume that the air resistance can be neglected and that the water starts from height 0.
We take the origin at the fountain, use cartezian coordinates, X and Y axes horizontal,
and consider a drop of water squirted with starting speed
$u = 16~ft/s$ in the plane (direction) of the X-axis, at angle $theta$ to the horizon.
Let the Z-axis be directed ...

Solution Summary

A fountain has jets that squirt water in all directions at a speed of 16 ft/sec from a central point as shown. The dotted curve shows the boundary between the "dry" region (which the fountain doesn't reach) and the "wet" region reached by the water spray. Find the shape of this dotted curve, giving an equation for it (y as a function of x). Ignore any air resistance, and assume that g is exactly 32 ft/sec2.

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