1. **State the problem:** We have a frisbee thrown with initial velocity 40 ft/s from a height of 5 ft. The height after $t$ seconds is given by the function $$h(t) = -16t^2 + 40t + 5.$$ We need to: a) Find the height of the frisbee at each second it is in the air. b) Determine how long the frisbee stays in the air. 2. **Formula and explanation:** The height function is a quadratic equation representing vertical motion under gravity (with acceleration $-32$ ft/s², hence the $-16t^2$ term). 3. **Part a) Find height at each second:** Calculate $h(t)$ for $t = 0, 1, 2, 3, \ldots$ until the frisbee hits the ground (height 0). - At $t=0$: $$h(0) = -16(0)^2 + 40(0) + 5 = 5$$ - At $t=1$: $$h(1) = -16(1)^2 + 40(1) + 5 = -16 + 40 + 5 = 29$$ - At $t=2$: $$h(2) = -16(2)^2 + 40(2) + 5 = -64 + 80 + 5 = 21$$ - At $t=3$: $$h(3) = -16(3)^2 + 40(3) + 5 = -144 + 120 + 5 = -19$$ (below ground, so frisbee is no longer in air) So the frisbee is in the air for about 2 seconds (since at 3 seconds height is negative). 4. **Part b) Find time in air:** Set height to zero and solve for $t$: $$0 = -16t^2 + 40t + 5$$ Use quadratic formula: $$t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a = -16$, $b = 40$, $c = 5$. Calculate discriminant: $$\Delta = 40^2 - 4(-16)(5) = 1600 + 320 = 1920$$ Square root: $$\sqrt{1920} = \sqrt{64 \times 30} = 8\sqrt{30}$$ Calculate roots: $$t = \frac{-40 \pm 8\sqrt{30}}{2(-16)} = \frac{-40 \pm 8\sqrt{30}}{-32}$$ Simplify numerator and denominator: $$t = \frac{\cancel{-40} \pm \cancel{8}\sqrt{30}}{\cancel{-32}} = \frac{-40 \pm 8\sqrt{30}}{-32}$$ Divide numerator and denominator by 8: $$t = \frac{-5 \pm \sqrt{30}}{-4}$$ Rewrite as: $$t = \frac{5 \mp \sqrt{30}}{4}$$ Two solutions: - $$t_1 = \frac{5 - \sqrt{30}}{4} \approx \frac{5 - 5.477}{4} = \frac{-0.477}{4} = -0.119$$ (negative time, discard) - $$t_2 = \frac{5 + \sqrt{30}}{4} \approx \frac{5 + 5.477}{4} = \frac{10.477}{4} = 2.619$$ seconds So the frisbee hits the ground after approximately 2.62 seconds. **Final answers:** - Heights at seconds 0,1,2 are 5 ft, 29 ft, and 21 ft respectively. - Frisbee is in the air for about 2.62 seconds.