Question

An arch is in the shape of a parabola. It has a span of 72 meters and a maximum height of 9 meters.

Answer 1

Answer:

Equation of the parabola:

$y=-\frac{1}{144}x^2+9$

The height of the arch 18 meters from the center is 6.75m

Explanation:

The arch feet are 72m apart and the origin half way between them. This means that the axis of symmetry (or the x.coordinate of the vertex) is x = 0

Since it's an arch, the parabola is concave down, with it's maximum at the vertex, y = 9. This means that the vertex is at (0, 9). Also, we can see that the y-intercept is y = 9

Finally, we know the two roots of the parabola: x = -36 and x = 36. This is because the points x = -36 and x = 36 are 72m apart, with the center at the original, as the problem says. SInce x = 36 is a root, this means that at that point the y value is 0.

With all this, we can try to find the general form of a parabola. The general form is:

$y=ax^2+bx+c$

c is the y-intercept. We know that c = 9

We can find the value of b, because we know the coordinates of the vertex. The x-coordinate of the vertex is:

$x_{vertex}=-\frac{b}{2a}$

SInce the x-coordinate of the vertex is x = 0

$0=-\frac{b}{2a}$

If we solve:

$b=0$

So far we have:

$y=ax^2+9$

Finally, to find a, we can use the point (36, 0) (one of the roots)

$0=a(36)^2+9$

And solve:

$\begin{gathered} 324a=-9 \\ . \\ a=-\frac{9}{1296}=-\frac{1}{144} \end{gathered}$

Thus, the equation of the arch is:

$y=-\frac{1}{144}x^2+9$

Evaluating this equation for x = 18, we can find the height of the arch:

$y=-\frac{1}{144}(18)^2+9=-\frac{9}{4}+9=-6.75$