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, as shown in the diagram. The paths of the two jets are modelled by y=-x^2-4 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.
(the diagram shows the two different height pools and two parabolas that both open downward. Both parabolas start at the same spot on the higher pool and land separate because ones bigger. I hope this helps, I don't know how to share pictures on here )
Please help and show steps, I tried and ended up with something wildly wrong.
The only interpretation I can make of this is that the water starts where the parabolas intersect once, and the lower pool is where they meet again. If so, then see the graphs at
https://www.wolframalpha.com/input/?i=solve+y%3D-x%5E2-4+%2C+y%3D-3x%5E2-15x
and note the red dots. If this is not how your picture looks, then I'm at a loss as to how to interpret things. The two pools could be anywhere on the parabolas.
To find the vertical height difference, d, between the two pools, we need to determine the y-coordinate at the point where the two parabolas intersect. This can be done by setting the equations equal to each other and solving for x.
Let's equate the two equations representing the paths of the water jets:
-x^2 - 4 = -3x^2 - 15x
Rearrange the equation:
0 = 2x^2 + 15x - 4
Now we need to solve this quadratic equation for x. There are different methods to solve quadratic equations, but we will use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
In this equation, a = 2, b = 15, and c = -4. Plugging in these values, we can solve for x:
x = (-15 ± √(15^2 - 4*2*(-4))) / 2*2
Simplifying the equation further:
x = (-15 ± √(225 + 32)) / 4
x = (-15 ± √257) / 4
Now, we have two possible values for x: x₁ and x₂.
x₁ = (-15 + √257) / 4 ≈ 0.55
x₂ = (-15 - √257) / 4 ≈ -7.3
Since the parabolas open downward, we can discard the negative value for x (-7.3) as it doesn't correspond to a valid point on the graph.
Now that we have the x-coordinate of the intersection point, we can substitute it into either equation to find the corresponding y-coordinate. Let's use the second equation:
y = -3x^2 - 15x
Substituting x = 0.55:
y = -3(0.55)^2 - 15(0.55)
y ≈ -0.9
The negative value means the intersection point is below the x-axis, which matches with the diagram where the lower pool is below the higher pool.
Finally, the vertical height difference, d, is given by the difference between the y-coordinate of the higher pool and the intersection point, so:
d = -4 - (-0.9) = -3.1
Rounding to the nearest hundredth, the vertical height difference, d, is approximately -3.10 feet.