Question

An archway in front of a school is in the shape of a parabola. The top of the arch is the vertex (0,0). The school seal is at the focus, 2.5 feet below the vertex, and the arch is 18 feet wide at the ground

Answer 1

a) The equation of the arch is
`y = -10\cdot x^(2)`.

b) The height of the arch from the ground is 810 feet.

Explanation:

Statement is incomplete. Complete statement is:

An archway in front of a school is in the shape of a parabola. The top of the arch is the vertex (0,0). The school seal is at the focus, 2.5 feet below the vertex, and the arch is 18 feet wide at the ground.

Write an equation that represents the arch.

What is the height from the top of the arch to the ground?

a) Write an equation that represents the arch.

According to the statement, the archway is a vertical downward parabola. From Analytical Geometry we know that parabolas are represented by the following equation in standard form:

`y-k = 4\cdot p\cdot (x-h)^(2)` (Eq. 1)

Where:

`p` - Distance between vertex and focus, measured in feet.

`x` - Independent variable, measured in feet.

`y` - Dependent variable, measured in feet.

`h`,
`k` - Coordinates of the vertex, measured in feet.

If we know that
`p = -2.5\,ft`,
`h = 0\,ft` and
`k = 0\,ft`, then the equation that represents the arch:

`y = -10\cdot x^(2)`

The equation of the arch is
`y = -10\cdot x^(2)`.

b) What is the height from the top of the arch to the ground?

Given that parabola is a symmetrical curve, we determine the vertical position at the base of the arch (
`x = 9\,ft`):

`y_(min) = -10\cdot (9\,ft)^(2)`

`y_(min) = -810\,ft`

Then, the height of the arch from the ground is given by this difference:

`h = 0\,ft-(-810\,ft)`

`h = 810\,ft`

The height of the arch from the ground is 810 feet.