To determine the equation of the parabolic shape of the gate, we need to find the appropriate parabolic equation given the dimensions: 80 feet wide and 25 feet tall.
We start by noting that the width of 80 feet means the gate extends 40 feet on either side of the origin, i.e., from [tex]\( x = -40 \)[/tex] to [tex]\( x = 40 \)[/tex].
The highest point of the arch (the vertex) will be at [tex]\( (0, 25) \)[/tex], since the gate is 25 feet tall.
The vertex form of a parabola is given by:
[tex]\[ y = a(x - h)^2 + k \][/tex]
Here, [tex]\( (h, k) \)[/tex] is the vertex. Substituting the coordinates of the vertex, we get:
[tex]\[ y = a(x - 0)^2 + 25 \][/tex]
[tex]\[ y = ax^2 + 25 \][/tex]
We also know that at the points where the gate meets the ground, [tex]\( y = 0 \)[/tex] (the base of the gate). This happens at [tex]\( x = \pm 40 \)[/tex]. So we use these points to find [tex]\( a \)[/tex]. Substitute [tex]\( x = 40 \)[/tex] and [tex]\( y = 0 \)[/tex] into the equation:
[tex]\[
0 = a(40)^2 + 25
\][/tex]
[tex]\[
0 = 1600a + 25
\][/tex]
To solve for [tex]\(a\)[/tex]:
[tex]\[
1600a = -25
\][/tex]
[tex]\[
a = -\frac{25}{1600}
\][/tex]
[tex]\[
a = -\frac{1}{64}
\][/tex]
Thus, the equation of the parabola in vertex form is:
[tex]\[ y = -\frac{1}{64}x^2 + 25 \][/tex]
To convert this into the standard parabolic form [tex]\( x^2 = 4p(y - k) \)[/tex]:
[tex]\[ y - 25 = -\frac{1}{64}x^2 \][/tex]
Multiplying both sides by [tex]\(-64\)[/tex] to eliminate the fraction:
[tex]\[ -64(y - 25) = x^2 \][/tex]
Or, rewriting it:
[tex]\[ x^2 = -64(y - 25) \][/tex]
Thus, the equation of the parabolic shape of the gate is:
[tex]\[ x^2 = -64(y - 25) \][/tex]
Therefore, the correct answer is:
C. [tex]\( x^2 = -64(y - 25) \)[/tex]
