Answer :
To find the equation of the parabolic shape of the gate, we need to follow these steps:
1. Define the coordinate system and the parabola form:
The arch is symmetric about the y-axis since it's the shape of a gate. Therefore, we can place the vertex of the parabola at the highest point of the arch, which is at [tex]\((0, 25)\)[/tex]. Given this, the standard form of the equation of the parabola that opens downwards is:
[tex]\[ x^2 = -a(y - 25) \][/tex]
where [tex]\((h, k) = (0, 25)\)[/tex] and [tex]\(-a\)[/tex] is a coefficient to be determined.
2. Determine the width of the arch:
The arch is 80 feet wide, so half the width is 40 feet. Therefore, the points where the parabola intersects the x-axis are [tex]\((40, 0)\)[/tex] and [tex]\((-40, 0)\)[/tex].
3. Substitute a point on the parabola to find the value of [tex]\(a\)[/tex]:
Substitute the point [tex]\((40, 0)\)[/tex] into the parabola equation [tex]\(x^2 = -a(y - 25)\)[/tex]:
[tex]\[ 40^2 = -a(0 - 25) \][/tex]
Simplify this equation:
[tex]\[ 1600 = 25a \][/tex]
Solving for [tex]\(a\)[/tex]:
[tex]\[ a = \frac{1600}{25} = 64 \][/tex]
4. Write the final equation:
Substitute [tex]\(a = 64\)[/tex] back into the parabola equation:
[tex]\[ x^2 = -64(y - 25) \][/tex]
Hence, the correct answer is:
[tex]\[ \boxed{x^2 = -64(y - 25)} \][/tex]
So, the correct answer is:
C. [tex]\(x^2 = -64(y - 25)\)[/tex]
1. Define the coordinate system and the parabola form:
The arch is symmetric about the y-axis since it's the shape of a gate. Therefore, we can place the vertex of the parabola at the highest point of the arch, which is at [tex]\((0, 25)\)[/tex]. Given this, the standard form of the equation of the parabola that opens downwards is:
[tex]\[ x^2 = -a(y - 25) \][/tex]
where [tex]\((h, k) = (0, 25)\)[/tex] and [tex]\(-a\)[/tex] is a coefficient to be determined.
2. Determine the width of the arch:
The arch is 80 feet wide, so half the width is 40 feet. Therefore, the points where the parabola intersects the x-axis are [tex]\((40, 0)\)[/tex] and [tex]\((-40, 0)\)[/tex].
3. Substitute a point on the parabola to find the value of [tex]\(a\)[/tex]:
Substitute the point [tex]\((40, 0)\)[/tex] into the parabola equation [tex]\(x^2 = -a(y - 25)\)[/tex]:
[tex]\[ 40^2 = -a(0 - 25) \][/tex]
Simplify this equation:
[tex]\[ 1600 = 25a \][/tex]
Solving for [tex]\(a\)[/tex]:
[tex]\[ a = \frac{1600}{25} = 64 \][/tex]
4. Write the final equation:
Substitute [tex]\(a = 64\)[/tex] back into the parabola equation:
[tex]\[ x^2 = -64(y - 25) \][/tex]
Hence, the correct answer is:
[tex]\[ \boxed{x^2 = -64(y - 25)} \][/tex]
So, the correct answer is:
C. [tex]\(x^2 = -64(y - 25)\)[/tex]
