A golt hall is hit with an initial velocity of 140 feet per second at an inclination of \( 45^{\circ} \) to the horizontal. \[ h(x)=\frac{-32 x^{2}}{140^{2}}+x \text {, } \] where \( x \) is the horizontal distance that the golf ball has fraveled Complete parts (a) through (g).

To find the horizontal distance that the golf ball has traveled, we need to determine the value of \( x \) when the golf ball hits the ground. Given: - Initial velocity = 140 feet per second - Inclination = \( 45^{\circ} \) - Equation of the trajectory: \( h(x) = \frac{-32x^2}{140^2} + x \) To find the horizontal distance, we need to find the value of \( x \) when the golf ball hits the ground, which means the height \( h(x) \) is equal to 0. Let's solve the equation \( h(x) = 0 \) to find the value of \( x \) when the golf ball hits the ground. Solve the equation by following steps: - step0: Solve for \(x\): \(\frac{-32x^{2}}{140^{2}}+x=0\) - step1: Divide the terms: \(-\frac{2x^{2}}{1225}+x=0\) - step2: Multiply both sides of the equation by LCD: \(\left(-\frac{2x^{2}}{1225}+x\right)\times 1225=0\times 1225\) - step3: Simplify the equation: \(-2x^{2}+1225x=0\) - step4: Factor the expression: \(-x\left(2x-1225\right)=0\) - step5: Separate into possible cases: \(\begin{align}&-x=0\\&2x-1225=0\end{align}\) - step6: Solve the equation: \(\begin{align}&x=0\\&x=\frac{1225}{2}\end{align}\) - step7: Rewrite: \(x_{1}=0,x_{2}=\frac{1225}{2}\) The solutions to the equation \( h(x) = 0 \) are \( x = 0 \) and \( x = \frac{1225}{2} \) or \( x = 612.5 \). Since the horizontal distance cannot be negative, the value of \( x \) when the golf ball hits the ground is \( x = 612.5 \) feet. Therefore, the horizontal distance that the golf ball has traveled when it hits the ground is 612.5 feet.