The arch support of a bridge can be modeled by $y = -0.0012x^2$, where x and y are measured in feet. Find the height and width of the arch.

Steps to Solve

1. Find the height of the arch

The height of the arch corresponds to the y-value when $x = 0$. Plug in $x = 0$ into the equation:

$y = -0.0012(0)^2 = 0$

However, from the picture, we can see that the arch goes below the x-axis, so this isn't the entire height. This equation describes the curve of the arch relative to its maximum point.

To find the height of the arch we need to find the y value where the arch meets the ground. We do that in the next steps.

2. Find the width of the arch

The width of the arch represents the distance between the two x-intercepts. To find the x-intercepts, set $y = -250$ because of the picture given in the problem

$-250 = -0.0012x^2$. Then solve for $x$.

3. Solve for x Divide both sides by $-0.0012$:

$x^2 = \frac{-250}{-0.0012} = 208333.33$

Take the square root of both sides:

$x = \pm\sqrt{208333.33} \approx \pm 456.435$

4. Calculate the width

The width is the difference between the two x-intercepts:

Width $= 456.435 - (-456.435) = 2 \times 456.435 \approx 912.87$ feet.

5. Calculate the height

Since the vertex is at y=0, and the arch extends to y=-250, the height is the absolute value of the difference.

Height $= |0 - (-250)| = 250$ feet.