To determine the maximum height of the Gateway Arch in St. Louis, we can analyze the given equation for the arch's shape:
[tex]\[ y = -\frac{2}{315}(x+315)(x-315) \][/tex]
This equation models a parabola, where [tex]\(x\)[/tex] and [tex]\(y\)[/tex] are measured in feet. The x-axis represents the ground.
### Step-by-Step Solution:
1. Identify the Form of the Equation:
The given equation is in the quadratic form where it can be rewritten as:
[tex]\[ y = -\frac{2}{315}(x^2 - 315^2) \][/tex]
This is recognized as a standard quadratic (or parabolic) equation in [tex]\(x\)[/tex].
2. Find the Vertex:
For a parabola given in the form [tex]\( y = a(x - h)^2 + k \)[/tex] or its simplified product form as in this case, the vertex and therefore the maximum or minimum point of the parabola is key. Given the symmetry, the vertex for a parabola of this form is found where [tex]\(x = 0\)[/tex].
3. Substitute [tex]\( x = 0 \)[/tex] into the Equation:
Substituting [tex]\( x = 0 \)[/tex] into the given equation will yield the [tex]\( y \)[/tex]-value at this point, which represents the height of the arch at its peak (vertex of the parabola).
[tex]\[ y = -\frac{2}{315} (0 + 315) (0 - 315) \][/tex]
4. Simplify the Expression:
Evaluate the expression step by step:
[tex]\[ y = -\frac{2}{315} (315)(-315) \][/tex]
[tex]\[ y = -\frac{2}{315} \times -99225 \][/tex]
[tex]\[ y = \frac{2}{315} \times 99225 \][/tex]
[tex]\[ y = \frac{198450}{315} \][/tex]
[tex]\[ y = 630 \][/tex]
5. Conclusion:
The height of the Gateway Arch at its peak is:
[tex]\[ y = 630 \text{ feet} \][/tex]
Thus, the maximum height of the Gateway Arch is 630 feet.
