1. **Problem Statement:** We analyze the quadratic height function of a bungee jumper given by $$h(t) = -0.5t^2 + v_0 t + h_0$$ with $$v_0 = 0$$ and $$h_0 = 210$$ meters. 2. **Domain and Range:** - The domain of $$h(t)$$ is the set of all possible times $$t$$ starting from 0 (jump time) until the jumper reaches the river (height 0). So, $$\text{Domain} = [0, t_{river}]$$. - The range is the set of heights from the minimum height (river level, 0) up to the maximum height reached. - Physically, the domain represents the time interval during which the jumper is in the air, and the range represents the possible heights above the river surface. 3. **Vertex of the Parabola:** - The vertex formula for $$h(t) = at^2 + bt + c$$ is $$t = -\frac{b}{2a}$$. - Here, $$a = -0.5$$, $$b = 0$$, so $$t = -\frac{0}{2(-0.5)} = 0$$. - The vertex is at $$t=0$$, $$h(0) = 210$$. - The vertex represents the initial height and the maximum height since the parabola opens downward. 4. **Maximum Height and Time:** - Since $$v_0=0$$, the jumper starts at maximum height $$210$$ meters at $$t=0$$. - The maximum height is $$210$$ meters at $$t=0$$ seconds. 5. **Time to Reach Height 11m:** - Solve $$h(t) = 11$$: $$-0.5t^2 + 210 = 11$$ $$-0.5t^2 = 11 - 210 = -199$$ $$t^2 = \frac{199}{0.5} = 398$$ $$t = \sqrt{398} \approx 19.95$$ seconds. - So, the jumper reaches 11 meters at approximately $$19.95$$ seconds. 6. **Height at 20 seconds:** - Calculate $$h(20) = -0.5(20)^2 + 210 = -0.5(400) + 210 = -200 + 210 = 10$$ meters. - This means after 20 seconds, the jumper is 10 meters above the river. 7. **Time to Touch the River:** - Solve $$h(t) = 0$$: $$-0.5t^2 + 210 = 0$$ $$-0.5t^2 = -210$$ $$t^2 = \frac{210}{0.5} = 420$$ $$t = \sqrt{420} \approx 20.49$$ seconds. - The jumper touches the river at approximately $$20.49$$ seconds. **Summary:** - Domain: $$[0, 20.49]$$ seconds. - Range: $$[0, 210]$$ meters. - Vertex: $$(0, 210)$$ maximum height. - Height 11m at $$t \approx 19.95$$ seconds. - Height at 20s is 10m. - Touches river at $$t \approx 20.49$$ seconds.