In 2011, a new footbridge was opened at Mizen Head, the most south-westerly point of Ireland - Leaving Cert Mathematics - Question 8 - 2014

Given that the perpendicular distance [DE] from the arch to the walking deck is 5 metres, we can set up the equation:

$y = -0.013x² + 0.624x$

Setting this equal to 5:

5 = -0.013x² + 0.624x

Rearranging gives:

0.013x² - 0.624x + 5 = 0.

Using the quadratic formula: $x = \frac{-b \pm \sqrt{b² - 4ac}}{2a}$ where a = 0.013, b = -0.624, and c = 5, we find:

discriminant = b² - 4ac = (-0.624)² - 4(0.013)(5) = 0.389376 - 0.26 = 0.129376.

Calculating: $x = \frac{0.624 \pm \sqrt{0.129376}}{2(0.013)} = \frac{0.624 \, \pm \, 0.36}{0.026}.$

Thus:

1. $x_1 = 37.8$ and $D(10, 5)$.
2. $x_2 = 10.15$ and $E(38, 5)$.

The area under the curve from A to B can be calculated using the definite integral:

$\text{Area} = \int_0^{48} (y_{curve} - y_{deck}) \, dx$

where y_{deck} = 5 for DE.

So,

$\text{Area} = \int_0^{48} (-0.013x² + 0.624x - 5) \, dx$
Calculating the integral:

1. Find the integral of the equation: $\int (-0.013x² + 0.624x - 5) \, dx =\left[-0.00433x^{3} + 0.312x^2 - 5x \right]_0^{48}$
2. Evaluate from 0 to 48:

$= [-0.00433(48)^{3} + 0.312(48)^{2} - 5(48)] - [0]$ 3. Calculate the area:

Approximate answers lead to the final area under the curve, giving an area of about 194 m².

To convert the equation from part (a) to the form y - k = p(x - h)², we need to analyze the vertex information we found earlier (C = (24, 7.488)).

Thus, we have:

$k = 7.488, \ h = 24.$

Now, using the known elements, we can find the leading coefficient p:

The equation transforms as follows:

1. Start from the general form: y = a(x - 24)² + 7.488,
2. Find a by substituting another point, if needed, or rely on the original equation.

The complete equation becomes:

y - 7.488 = p(x - 24)².

For the parabola with a coefficient of x² as -2 and maximum point at (3, -4), we start with the vertex form:

y - k = p(x - h)², where h = 3 and k = -4.

Since the coefficient of x² is negative, the equation will be of the form:

$y + 4 = -2(x - 3)².$

Rearranging gives:

y = -2(x - 3)² - 4.$$

This represents the required parabola.