1. **State the problem:** We have the height of a golf ball modeled by the equation $$h = -0.005d^2 + 1.4d + 2$$ where $h$ is the height in meters and $d$ is the horizontal distance in meters. We need to find: a) The horizontal distance $d$ when the ball hits the ground, i.e., when $h=0$. b) The range of $d$ for which the height $h$ is greater than 50 meters. --- 2. **Formula and rules:** This is a quadratic equation in $d$. To find when the ball hits the ground, solve for $d$ when $h=0$: $$0 = -0.005d^2 + 1.4d + 2$$ To find when $h > 50$, solve: $$-0.005d^2 + 1.4d + 2 > 50$$ --- 3. **Solve part (a):** Set $h=0$: $$-0.005d^2 + 1.4d + 2 = 0$$ Multiply both sides by $-1$ to simplify: $$0.005d^2 - 1.4d - 2 = 0$$ Use the quadratic formula: $$d = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=0.005$, $b=-1.4$, $c=-2$. Calculate discriminant: $$\Delta = (-1.4)^2 - 4(0.005)(-2) = 1.96 + 0.04 = 2.0$$ Calculate roots: $$d = \frac{-(-1.4) \pm \sqrt{2.0}}{2 \times 0.005} = \frac{1.4 \pm 1.4142}{0.01}$$ Two solutions: $$d_1 = \frac{1.4 - 1.4142}{0.01} = \frac{-0.0142}{0.01} = -1.42$$ (discard negative distance) $$d_2 = \frac{1.4 + 1.4142}{0.01} = \frac{2.8142}{0.01} = 281.42$$ Rounded to one decimal place: $$d = 281.4$$ meters --- 4. **Solve part (b):** Find $d$ such that: $$-0.005d^2 + 1.4d + 2 > 50$$ Rewrite: $$-0.005d^2 + 1.4d + 2 - 50 > 0$$ $$-0.005d^2 + 1.4d - 48 > 0$$ Multiply both sides by $-1$ (remember to reverse inequality): $$0.005d^2 - 1.4d + 48 < 0$$ Use quadratic formula with $a=0.005$, $b=-1.4$, $c=48$: Calculate discriminant: $$\Delta = (-1.4)^2 - 4(0.005)(48) = 1.96 - 0.96 = 1.0$$ Calculate roots: $$d = \frac{1.4 \pm 1.0}{2 \times 0.005} = \frac{1.4 \pm 1.0}{0.01}$$ Roots: $$d_1 = \frac{1.4 - 1.0}{0.01} = \frac{0.4}{0.01} = 40$$ $$d_2 = \frac{1.4 + 1.0}{0.01} = \frac{2.4}{0.01} = 240$$ Since the parabola opens downward (coefficient of $d^2$ is negative), $h > 50$ between the roots: $$40 < d < 240$$ meters --- **Final answers:** a) The ball hits the ground at approximately $281.4$ meters. b) The height of the ball is greater than 50 meters when the horizontal distance is between $40$ meters and $240$ meters.