A parabolic bridge over a river is 30.0 m wide and 12.0 m high. Find the equation, in factored form, that represents the parabolic arch of the bridge.
struggling please help with this
since the bridge is 30m wide, its highest point is at (0,12)
set it up as y = 12 - ax^2
where y(15) = 0
or, you could start with y = a(x-15)(x+15)
where y(0) = 12
You're given the y-coordinate of the vertex (__, 12), but the x-coordinate is missing. However, that can easily be calculated by determining the
midpoint of 0 and 30 (15). This is the best way I can visually explain it (imagine the arch being drawn).
12 |
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0 15 30
This can be expressed in factored form:
y= a(x-0)(x-30)
y= ax(x-30)
Since we have our vertex/another point, you can calculate further to determine the value of a.
12= a(15)(15-30)
12= a(15)(-15)
12= -225a
-12/225 = a
The complete equation is now: y= -12/225(x)(x-30)
The visual representation kind of messed up...
12|
. |
. |_________________
. 0. 15. 30
To find the equation that represents the parabolic arch of the bridge, we can start by considering the vertex form of a parabola equation:
y = a(x - h)^2 + k
Where (h, k) represents the coordinates of the vertex of the parabola.
In this case, the width of the bridge is given as 30.0 m, which means that the parabolic arch is symmetric about the y-axis. Therefore, the vertex lies on the y-axis, and the x-coordinate of the vertex is 0.
Given that the height of the bridge is 12.0 m, we can conclude that the y-coordinate of the vertex is 12.0. Hence, the vertex of the parabolic arch is (0, 12).
Now, we have the values of h and k, which we can substitute into the vertex form equation to obtain the equation in factored form:
y = a(x - 0)^2 + 12
Simplifying further, we get:
y = ax^2 + 12
Thus, the equation, in factored form that represents the parabolic arch of the bridge, is y = ax^2 + 12.
Note that we don't have enough information to determine the value of 'a' in this case, as we would need more data or additional constraints to find its exact value.