A parabolic arch in a school corridor is wide at the base

MATHEMATICS · Multi-part · GCSE · Mathematics · Graphs · Graphs

Problem

A parabolic arch in a school corridor is

12\,\text{m}

wide at the base. The highest point of the arch is

5\,\text{m}

above the ground and is located at the midpoint of the base. Let the midpoint of the base be taken as the origin

O(0,0)

of a coordinate system, with the

x

-axis along the ground and the

y

-axis vertically upward. The height of the arch at a horizontal distance

x

metres from

O

is denoted by

h(x)

. (a) Express

h(x)

in the form

ax^{2}+bx+c

by using the information given about the arch. (b) Determine the numerical values of

a,\,b

and

c

and hence write the explicit expression for

h(x)

. (c) Using the obtained model, calculate the height of the arch at a point

3\,\text{m}

away from the midpoint along the ground.