a The equation below models the shape of an arch decorated with balloons.
y=- 0.5x^2+4.5x
In this equation, x represents the floor and y is the height of the arch. Hence, we need to factor the right-hand side of the equation.
This expression represents the height in factored form.
b The height y of the arc is zero at the points where the arch touches the ground. In Part A, we have written expression for the height of the arch in factored form.
- 0.5x(x-9)
We need to solve for x when this expression is equal to zero. To do so, we will apply the Zero Product Property.
The difference between these points on the x-axis is 9, so they are 9 feet away from each other.
c We will use the maximum feature of our graphing calculator. Before that, let's first draw the function on the calculator. Push the Y= button and type the equation in the first row.
Now we can push GRAPH to draw them.
We are not able to see the entire graph. We need to change the viewing window.
Next, we can find the local maximum. We need to push 2nd, then TRACE, and choose the maximum option.
When using the maximum feature, we are prompted to choose left and right bounds and then provide the calculator with a best guess as to where the maximum might be.
The maximum is at 10.125. Therefore, the highest point of the arch is 10.125 ft.
