Let's graph the quadratic function representing the height of the ball and the horizontal line representing the height of the goal.
The ball will be able to enter the goal when it is above ground and the height is less than 8 feet.
0≤ -0.1t^2+2.4t+1.5<8
Graphically, this means that we need to find the t-values for which the blue parabola is below the red line.
To find the solution, we need to find the intersection points of the line and the parabola and the axes intercepts of the parabola.
Let's use a calculator to find these points.
We begin by pushing the Y= button and typing the equation in the first two rows.
To see the graphs, you will need to adjust the window.
Push WINDOW, change the settings, and push GRAPH.
Next, to find the intersection, push 2nd and TRACE. From this menu, choose intersect.
The calculator will prompt you to choose the first and second curve and to provide the calculator with a best guess of where the intersection might be.
You will need to repeat the process twice to find both intersection points.
To find the horizontal axis intercept, push 2nd and TRACE again. This time choose zero from the menu.
The calculator will prompt you to choose a left and right bound and to provide the calculator with a best guess of where the zero might be.
Note that the calculator uses a numerical method, so the y-coordinate of the axis intercept is only approximately 0.
Let's copy the results on our graph.
The ball will be able to enter the goal if the player is close to the goal or if the player is far away from the goal.
0≤ t<3.11 or 20.88
