I have learned the formula y = a(x-s)(x-t).
bobpursley
answered
y=a(x)(x-48)
and at x=24 (half way across), y=12
12=a(24)(-24)
solve for a ( I get in my head -1/48)
Explain Bot
answered
Given that the dome has a parabolic shape and the equation y = a(x - s)(x - t) is factored form, we can use the information provided to determine the values of s and t.
From the measurements given, we know that the parabolic dome has a width of 48 m, which means the distance from one side to the other is 48 m. In the factored form of the equation, the difference between s and t represents the width of the parabola.
Since the width is 48 m, we can write:
t - s = 48
We also know that the parabola rises 12 m at its center. The x-coordinate of the vertex represents the center of the parabola, and at this point, the y-coordinate is the maximum or minimum value of the parabola. In this case, the center is the vertex, and the maximum value is 12 m.
Using the vertex form of a parabola, which is y = a(x - h)^2 + k, we can determine the vertex of the parabola by using the values from the given measurements:
h = (s + t)/2 = (s + (s + 48))/2 = (2s + 48)/2 = s + 24
k = 12
Substituting these values into the vertex form, we have:
y = a(x - (s + 24))^2 + 12
Simplifying this equation, we get:
y = a(x - s - 24)^2 + 12
Now we have the parabolic equation in vertex form, and we can determine the values of a, s, and t by comparing this equation to the factored form equation y = a(x - s)(x - t):
s + 24 = t
12 = 24^2a
From the first equation, we can solve for t:
t = s + 24
Substituting this back in the factored form equation, we have:
48 = 48^2a
Now we can solve for a:
48^2a = 48
a = 1/(48^2)
Finally, substituting the values of a, s, and t back into the factored form equation, we get the final quadratic equation that models the shape of the parabola:
y = 1/(48^2)(x - s)(x - t)
