SOLUTION: Hi, this question is suppose to be solved using systems of equations but I'm not sure how to solve it. As part of a fountain display, two jets of water travel from the surface of

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Question 1092324: Hi, this question is suppose to be solved using systems of equations but I'm not sure how to solve it.
As part of a fountain display, two jets of water travel from the surface of an upper pool and
meet at the surface of a lower pool. The paths of the two jets are
modelled by y=-x^2-4x and y=-3x^2-15x , where x and y are measured in feet. What is
the vertical height difference, d, of the pools? Round the answer to the nearest hundredth.
Thank you for your help
Found 2 solutions by ikleyn, rothauserc:
Answer by ikleyn(53765)   (Show Source): You can put this solution on YOUR website!
.


Plot y = (red) and y = (green) 

This plot is to illustrate the problem.


You have two jets released from the origin (0,0).

They intersect somewhere. You need to find the intersection point.


For it, since "y"-s are equal, you can equate the right sides:


-x^2-4x = -3x^2-15x.


Which implies

2x^2 + 11x = 0  ====>  x*(2x + 11) = 0. ====>

====>  the only interesting root is 2x + 11 = 0,  i.e.  x = .


Then y =  = -8.25.


Answer.  The vertical height difference, d, of the pools is 8.25 ft.

Solved.



Answer by rothauserc(4718)   (Show Source): You can put this solution on YOUR website!
both of these equations are parabolas that curve downward
:
we want the intersection point/s of the two parabolas
:
-x^2 -4x = -3x^2 -15x
:
2x^2 +11x = 0
:
x^2 +11x/2 = 0
:
complete the square
:
(x +11/4)^2 = 121/16
:
take square root of both sides of =
:
x +11/4 = + or - 11/4
:
x = 0 or x = -11/2
:
we have two intersection points
:
x = 0, y = 0 and x = -11/2, y = -33/4
:
the point (0,0) is where the two jets begin at the upper pool(y = 0 at this time)
:
y = -33/4 is the height of the upper pool above the lower pool where the jets meet
:
**************************************************************************
33/4 = 8.25
:
the vertical height difference is 8.25 feet
:
here is the graph of the two jets, y = -x^2 -4x(red line), y = -3x^2 -15x(green line)
:

:
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