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Parabola bridge example

Question:

The figure shows a parabola bridge. The height from the top of the arch to the water level is 8 meters. The water level inside the arch bridge is 64 meters. When the water level rise one meter, how many waters wide is the water level inside the arch bridge?

how to solve a parabola bridge problem?

Solution:

Draw a line passing the point C and parallel to AB, name this line as x-axis.

Draw a line passing the point C and perpendicular to the x-axis, name this line as y-axis.

Name the point C as point O, which is the origin.

In this XOY coordinate plane, because the vertex of the curve is in the origin, the curve is symmetry to the negative y-axis and the curve is open down. Because the curve is a parabola, so, the curve has the equation: x2 = -2py, (p > 0).

Now, we will determine the value of p. The line AB is the water level. The point B is on the parabola. So, the point B satisfy the parabola equation. The point B has the coordinate (32, -8). We substitute the coordinate of point B into the parabola equation x2 = -2py to determine the value of p.

x2 = -2py
322 = - 2p (-8)
(32) (32) = 16 p
p = 64

The parabola has the equation: x2 = -2py = -2(64)y = -128 y

When the water level rises one meter, water goes to EF line. Point F has the coordinate F (x, -7).

Now, we substitute the coordinate of the point F into the parabola equation to get the x-coordinate of the point F.

x2 = -128y = -128 × (-7) = 896
x = +- square root of 896 = +- 29.93

Because the x-coordinate of the point F is a positive number, so, we drop the negative x-value.

 x = 29.93 = 30.
EF = 2x = 2 (30) = 60 meter

Therefore, the length of EF is 60 meters. When the water level rises one meter, the water width under the arch bridge is 60 meters. Watch the video for more details.