Chapter 1.2, Problem 52PS

Solution

To find: The width of the arch.

The width of the arch is 105.

Given information:

The arch of the bridge forms a parabola with the equation $y=-0.016{(x-52.5)}^2+45$ , where $x$ is the distance from the left end of the arch and $y$ is the distance from the base of the arch.

Concept used:

Vertex form a quadratic function is given as $y=a{\left(x-h\right)}^2+k$ , where $\left(h,k\right)$ is the vertex of the quadratic function.

The vertex of the quadratic function represents the maximum/minimum point of the function.

If $a$ positive, the vertex is the minimum point and if $a$ is negative, the vertex is the maximum point of the function.

The equation $x=h$ divides the parabola into two equal halves and is known as the equation of symmetry of a parabola.

Calculation:

Consider the equation of the parabola, $y=-0.016{(x-52.5)}^2+45$ .

The equation of symmetry for this parabola is $x=52.5$ . That mean, the half width of the arch from the left end is $52.5$ . So, the total width of the arch is $2\times 52.5=105$ .

Conclusion:

The width of the arch is 105.