Let's graph the quadratic function representing the arch and the horizontal line representing the height of 7 feet.
The arch is higher than 7 feet when f(x)≥ 7.
- x^2+6x+1≥ 7
Graphically, this means that we need to find the x-values for which the blue parabola is above the red line.
To find the solution, we need to find the intersection points of the line and 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.
Let's copy the results on our graph.
The arch is taller than 7 feet for x-values between the two x-intercepts.
1.27≤ x≤ 4.73
Extra
How far are these points from the side of the arch?
We've found the answer to the question the problem is asking, but let's think about it more.
To see how far these points are from the sides of the arch, we also need the point where the arch meets the ground.
To find the horizontal axis intercept of the graph, 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 add this result to our graph.
Using the axis intercept and the intersection points we found earlier, we can find the distance from the side where the arch is taller than 7 feet.
2
&Minimum distance:&& 6.16-4.73=1.43feet
&Maximum distance:&& 6.16-1.27=4.89feet
