Question

A parabolic archway is 25 meters high at the vertex. At a height of 23 meters, the width of the archway is 16 meters, as shown in the figure.

How wide (in m) is the archway at ground level? (Round your answer to the nearest whole number.)

Answer 1

We can use the fact that the shape of a parabolic arch follows the equation:

y = a(x - h)^2 + k

where (h, k) is the vertex of the parabola and "a" determines the shape of the curve.

In this case, we know that the vertex of the arch is at (0, 25), so we have:

y = a(x - 0)^2 + 25

We also know that when y = 23, the width of the arch is 16. We can use this information to solve for "a":

23 = a(x - 0)^2 + 25

Subtracting 25 from both sides, we get:

-2 = a(x - 0)^2

Dividing both sides by -2, we get:

a = -1 / (x - 0)^2

We can now use this value of "a" to find the equation of the parabolic arch. We know that when y = 0 (ground level), the width of the arch will be its maximum value. To find this value, we can use the fact that the x-coordinate of the maximum value of a parabola is given by:

x = h

where (h, k) is the vertex. In this case, h = 0 and k = 25, so we have:

x = 0

Substituting this into our equation for "a", we get:

a = -1 / (0 - 0)^2 = -1/0 (which is undefined)

This means that the shape of the arch is not a parabola, since "a" cannot be defined. Therefore, we cannot use the equation of a parabolic arch to find the width of the arch at ground level.