SOLUTION: a fountain shoots out water that follows a parabolic curve. water comes out of a port on the ground and goes in another port on the ground as well. if the ports are 15 feet away a
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-> SOLUTION: a fountain shoots out water that follows a parabolic curve. water comes out of a port on the ground and goes in another port on the ground as well. if the ports are 15 feet away a
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Question 949634: a fountain shoots out water that follows a parabolic curve. water comes out of a port on the ground and goes in another port on the ground as well. if the ports are 15 feet away and the water peaks at 12 feet from the ground, what is the horizontal distance of the water arc 4 feet from the ground? Answer by Theo(13342) (Show Source):
You can put this solution on YOUR website! you have a parabola that has a height of 12 feet and the distance between its ports is 15 feet.
if the axis of symmetry of the parabola is at x = 0, then the vertex will be at (0,12).
if the y-axis is the ground, then the ports will be 15 feet apart when y = 0.
the distance from each point to the axis of symmetry at x = 0 will be 7.5 feet.
you will have zero crossings of the parabola at x = -7.5 and x = 7.5
the equation would be a * (x-7.5) * (x+7.5) = 0
that simplifies to a * (x^2 -56.25) = 0
the general equation would therefore be y = a * (x^2 - 56.25)
when y = 12, the parabola will be at it's peak.
this happens when x = 0.
equation becomes 12 = a * (-56.25)
a is therefore equal to 12/-56.25 which is equal to -.21333333333 which can also be expressed as a fraction that is equal to -16/75.
equation of y = a * (x^2 - 56.25) becomes y = -16/75 * (x^2 - 56.25) which then becomes:
y = -16/75*x^2 + 12
that's the equation of your parabola.
when the water is at the 4 foot high mark, the parabola intersects with the line y = 4.
you get a system of equations where:
first equation is y = -16/75 * x^2 + 12
second equation is y = 4
replace y in the first equation with its equivalent value of 4 from the second equation and solve for x to get:
4 = -16/75 * x^2 + 12
subtract 12 from both sides of this equation to get -8 = -16/75 * x^2
divide both sides of this equation by -16.75 to get -8 / (-16/75) = x^2
solve for x to get x = plus or minus sqrt(-8/(-16/75)).
this results in x = plus or minus 6.123724357.
when x = plus or minus 6.123724357, the value of y will be equal to 4.
the distance between x = 6.123724357 and x = -6.123724357 is equal to 12.24744871 feet.
here's a graph of your equation with all the important intersections shown.
note that 6.1237... rounds to 6.124 as shown on the graph.
