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 \( y=-2 x^{2}+6.5 x+2 \). 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.

To find how far from the nozzle the water will hit the surface of the pond, we need to determine where the height \( y \) is equal to 0 (the surface of the pond). We start with the given equation: \[ y = -2x^2 + 6.5x + 2 \] Setting \( y = 0 \): \[ 0 = -2x^2 + 6.5x + 2 \] This leads us to solve the quadratic equation: \[ 2x^2 - 6.5x - 2 = 0 \] Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 2 \), \( b = -6.5 \), and \( c = -2 \): 1. Calculate the discriminant: \[ b^2 - 4ac = (-6.5)^2 - 4 \cdot 2 \cdot (-2) = 42.25 + 16 = 58.25 \] 2. Now applying the formula: \[ x = \frac{-(-6.5) \pm \sqrt{58.25}}{2 \cdot 2} = \frac{6.5 \pm 7.645}{4} \] This gives us two potential solutions: 1. \[ x_1 = \frac{6.5 + 7.645}{4} = \frac{14.145}{4} \approx 3.53625 \] 2. \[ x_2 = \frac{6.5 - 7.645}{4} = \frac{-1.145}{4} \approx -0.28625 \] Since a negative distance doesn’t make sense in this context, we discard \( x_2 \). Thus, rounding \( x_1 \) to the nearest hundredth, we find: **The water will hit the surface of the pond approximately 3.54 feet away from the nozzle.**