The problem involves modeling the facade of a building as part of a parabola. We are given:

• The height of the facade (45 meters) and
• The width of the facade (60 meters).

We can use this information to solve the following tasks:

a) Determine the intersections with the x-axis and the y-axis.

Since the facade's shape is modeled as part of a parabola, with its axis of symmetry along the y-axis, we can assume the general form of the parabola is:

$f(x) = ax^2 + bx + c$

However, because the parabola is symmetric along the y-axis, $b = 0$, and the equation simplifies to:

$f(x) = ax^2 + c$

The width of the facade is 60 meters, meaning the parabola spans from $x = -30$ to $x = 30$ (since it's symmetric). The height at the vertex, which corresponds to $x = 0$, is 45 meters. Therefore:

• The vertex is at $(0, 45)$, which gives $f(0) = 45$, so $c = 45$.
• The parabola intersects the x-axis at $x = -30$ and $x = 30$. This means that $f(-30) = 0$ and $f(30) = 0$.

Now, we substitute into the equation $f(x) = ax^2 + 45$, using the fact that $f(30) = 0$:

$0 = a(30)^2 + 45$ $0 = 900a + 45$ $900a = -45$ $a = -\frac{45}{900} = -\frac{1}{20}$

Thus, the equation of the parabola is:

$f(x) = -\frac{1}{20}x^2 + 45$

b) Determine the function for $f$.

From the calculations above, the function for $f(x)$ is:

$f(x) = -\frac{1}{20}x^2 + 45$

Summary:

• The parabola intersects the x-axis at $x = -30$ and $x = 30$, and the y-axis at $y = 45$.
• The equation of the parabola is $f(x) = -\frac{1}{20}x^2 + 45$.

Would you like further details or clarifications? Here are 5 related questions:

1. How would the equation change if the height of the building were doubled?
2. Can this model be used to calculate the area under the curve of the facade?
3. How would the parabola change if the building was 80 meters wide instead of 60 meters?
4. What is the geometric significance of the negative sign in $a = -\frac{1}{20}$?
5. How could we use this model to determine the volume of a similarly shaped 3D structure?

Tip: Always identify the vertex of the parabola first when dealing with symmetrical shapes like this one.