Question

approximating a solution from a graph
the function shown models the horizontal and vertical distances, in feet, traveled by a person on a roller coaster. what horizontal distance is traveled to first reach a height of 200 feet? use the drop - down menus to complete the statements.
the person first reaches a height of 200 feet after traveling a horizontal distance between
feet.
the horizontal distance traveled to reach a height of 200 feet is approximately
feet.
the graph has the equation $y = - 0.0002x^{3}-0.09x^{2}+10x + 16$ and a coordinate system with x - axis marked at 50, 100 and y - axis marked at 100, 200, 300, and the curve of the function plotted.

Explanation:

Step1: Analyze the graph's grid

The x - axis has grid lines. Let's assume each grid square on the x - axis has a width. From the graph, we can see that when y = 200 (the horizontal line at y = 200), we need to find the x - value (horizontal distance) where the curve first intersects y = 200. Looking at the x - axis, the grid lines are at intervals. Let's consider the x - values. The first intersection (first time reaching y = 200) occurs between x = 10 and x = 20? Wait, no, looking at the graph, the x - axis has marks at 50 and 100. Wait, maybe each grid square is 10 units? Let's check the y - axis: 100, 200, 300. The curve starts below the x - axis, rises, crosses y = 200. Let's see the x - values. When x = 10, let's plug into the function \(y=- 0.0002x^{3}-0.09x^{2}+10x + 16\). For x = 10: \(y=-0.0002\times1000-0.09\times100 + 10\times10+16=-0.2 - 9+100 + 16 = 106.8\). For x = 20: \(y=-0.0002\times8000-0.09\times400+10\times20 + 16=-1.6-36 + 200+16 = 178.4\). For x = 30: \(y=-0.0002\times27000-0.09\times900+10\times30 + 16=-5.4-81+300 + 16 = 229.6\). Wait, but we need y = 200. So between x = 20 and x = 30? Wait, no, maybe my grid assumption is wrong. Wait the graph has x - axis with 50 and 100 marked. Let's look at the graph: the curve rises, reaches a peak, then falls. The first time it reaches y = 200 is between x = 10 and x = 30? Wait, maybe the grid squares are 10 units. Let's see, when x = 20, y is less than 200 (178.4), when x = 30, y is more than 200 (229.6). Wait, but the problem says "between" what? Wait maybe the first drop - down is between 10 and 30? No, maybe the options are between 10 and 30, 30 and 50, etc. Wait, maybe I made a mistake. Wait the function is a cubic, but the graph looks like a parabola (maybe the cubic term is small). Let's look at the graph: the curve starts, crosses the y - axis, rises, reaches a peak around x = 50, then falls. The line y = 200 intersects the curve twice: once on the way up, once on the way down. The first intersection (on the way up) is between x = 10 and x = 30? Wait, no, when x = 20, y = 178.4; x = 30, y = 229.6. So between x = 20 and x = 30? But maybe the grid is such that each square is 10 units. Alternatively, maybe the first drop - down is between 10 and 30, and the approximate value is 25? Wait, no, let's check the graph again. The graph shows that the first time y = 200 is between x = 10 and x = 30? Wait, maybe the options for the first drop - down are 10 and 30, 30 and 50, etc. Wait, maybe I miscalculated. Let's try x = 25: \(y=-0.0002\times15625-0.09\times625+10\times25 + 16=-3.125-56.25 + 250+16 = 206.625\). Oh, that's close to 200. Wait, x = 24: \(y=-0.0002\times13824-0.09\times576+10\times24 + 16=-2.7648-51.84+240 + 16 = 201.3952\). Oh, that's very close to 200. So x≈24. So the first distance between is 20 and 30 (or 10 and 30), and the approximate value is 25 (or 24). Wait, maybe the drop - down for the first part is between 10 and 30 (or 20 and 30), and the second part is approximately 25 (or 24). But let's look at the graph: the x - axis has marks at 50 and 100, and the grid lines between them. So each grid square is 10 units (since 50 to 100 is 50 units, 5 squares? No, 50 to 100 is 50, so each square is 10 units). So the first time y = 200 is between x = 20 and x = 30 (since at x = 20, y is ~178, at x = 30, y is ~230). So the first drop - down: between 20 and 30? Wait, maybe the options are 10 and 30, 30 and 50, etc. Wait, maybe the problem's drop - down for the first part is between 10 and 30 (or 20 and 30), and the approximate value is 25 (or 24). But let's check the function at x = 25: y = 206.6, which is close to 200. At x = 24, y≈201.4, which is very close. So the horizontal distance traveled to reach 200 feet is approximately 25 (or 24) feet. And the first distance is between 20 and 30 (or 10 and 30) feet.

Step2: Determine the interval and approximation

From the function evaluation, when x = 20, y = 178.4 (less than 200), when x = 30, y = 229.6 (more than 200). So the first time reaching y = 200 is between 20 and 30 feet. Then, to approximate, we can use linear approximation. The difference between x = 20 (y = 178.4) and x = 30 (y = 229.6). We need y = 200. The difference in y: 200 - 178.4 = 21.6. The total difference in y: 229.6 - 178.4 = 51.2. The fraction is \(\frac{21.6}{51.2}=\frac{27}{64}\approx0.422\). So the x - value is 20+0.422×10≈24.22, so approximately 25 (or 24) feet.

Answer:

The person first reaches a height of 200 feet after traveling a horizontal distance between \(\boldsymbol{20}\) and \(\boldsymbol{30}\) feet (or other appropriate interval based on graph/function). The horizontal distance traveled to reach a height of 200 feet is approximately \(\boldsymbol{25}\) (or \(\boldsymbol{24}\)) feet. (Note: The exact interval and approximation may vary slightly based on graph interpretation, but typical answers for this problem are between 20 - 30 and approximately 25.)