Question

austin jumped off a cliff into a river. the following graph of a quadratic equation represents austins height h, in meters, above the river t seconds after jumping. which of the following values of t is a solution for when austin landed in the river?
5
1
none of the above
0
3

Explanation:

Step1: Understand the problem

We need to find the time \( t \) when Austin lands in the river, which means his height \( h = 0 \) (since the river is at height 0 meters). So we need to find the positive root of the quadratic graph (because time can't be negative) where \( h(t)=0 \).

Step2: Analyze the graph

The graph of the quadratic equation (a parabola) intersects the \( t \)-axis (where \( h = 0 \)) at two points. One is at a negative \( t \) (which we can ignore because time can't be negative before jumping) and the other is at a positive \( t \). From the graph, we can see that the positive intersection with the \( t \)-axis is around \( t = 3 \) (looking at the grid, the parabola crosses the \( t \)-axis at \( t\approx3 \), since between \( t = 0 \) and \( t = 5 \), and closer to 3? Wait, actually, looking at the options (even though the options are a bit unclear, but from the context, the options seem to be 5, 0, 1, 3, or "None of the above"? Wait, the user's image has options at the bottom: 5, 0, 1, 3, and "None of the above". Wait, no, the bottom part: "5", "0", "1", "3", "None of the above". Wait, let's re-examine. The graph: the parabola opens downward? Wait, no, the parabola here: when \( t = 0 \), \( h \) is positive (since he jumps from a cliff, so initial height is positive). Then it goes up, then down. Wait, no, the graph: the y-axis is \( h \) (meters), x-axis is \( t \) (seconds). The parabola crosses the t-axis at two points: one negative (left of origin) and one positive (right of origin). The positive crossing: looking at the grid, each grid square is, say, 1 unit. So from the graph, the positive root (where \( h = 0 \)) is at \( t = 3 \)? Wait, the options given (from the bottom of the image: 5, 0, 1, 3, None of the above). Wait, let's check: when \( t = 3 \), does the graph cross the t-axis? Let's see the graph: the parabola goes from negative t (left) crossing at negative t, then up, then down, crossing the t-axis at positive t. Let's see the x-axis (t-axis) labels: -15, -10, -5, 0, 5, 10, 15. The parabola crosses the t-axis at, say, around \( t = 3 \)? Wait, no, maybe the options: the options are 5, 0, 1, 3, None of the above. Wait, when \( t = 3 \), is that where he lands? Let's think again. The height is 0 when he lands. So we need to find the t where h(t)=0. The graph: the parabola intersects the t-axis at t ≈ 3 (since between t=0 and t=5, and the graph crosses the t-axis at t=3? Wait, maybe the correct answer is 3. Wait, the options: 5, 0, 1, 3, None of the above. So let's check each option:

• \( t = 5 \): At t=5, is h=0? From the graph, at t=5, the parabola is at the bottom (h=-15?), no, wait the graph: the parabola goes down to t=5? No, the right end of the parabola is at t=5? Wait, maybe the graph's positive root is at t=3. So the solution is t=3.

Answer:

3 (assuming the option is "3" as the correct time when he lands in the river, since that's when h(t)=0, the positive root of the quadratic representing his height over time.)