Itranscript
An outdoor fountain is to be installed in the center of a pond. The water jets will create a parabolic stream of water modeled by the equation $y=-2 x^{2}+8 x+1$. where $y$ is the height of the jet of water and $x$ is the horizontal distance of the jet of water from the water nozzle, both in feet. Determine how far from the nozzle the water will hit the surface of the pond. Round your answer to the nearest hundredth, if necessary.
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Step 1 ：The water will hit the surface of the pond when the height of the jet of water is zero, i.e., when \(y=0\). So, we need to solve the equation \(-2x^2 + 8x + 1 = 0\) for \(x\).

Step 2 ：This is a quadratic equation, and we can solve it using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = -2\), \(b = 8\), and \(c = 1\).

Step 3 ：However, since the distance cannot be negative, we only consider the positive root.

Step 4 ：By substituting the values of \(a\), \(b\), and \(c\) into the quadratic formula, we get two roots \(x1 = 4.121320343559642\) and \(x2 = -0.12132034355964239\).

Step 5 ：Since the distance cannot be negative, we discard \(x2\) and take \(x1\) as the distance from the nozzle to where the water hits the surface of the pond.

Step 6 ：Rounding to the nearest hundredth, we get \(x = 4.12\) feet.

Step 7 ：Final Answer: The water will hit the surface of the pond approximately \(\boxed{4.12}\) feet from the nozzle.