The cables of a suspension bridge are in the shape of a parabola, as shown in the figure. The towers supporting the cable are 300 feet apart and 20 feet high. If the cables touch the road surface midway between the towers, what is the height of the cable at a point 75 feet from the center of the bridge?
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To find the height of the cable at a point 75 feet from the center of the bridge, we need to find the equation of the parabola that represents the shape of the cables.
Since the cables touch the road surface midway between the towers and the towers are 300 feet apart, the center of the bridge is 150 feet from each tower.
Let's choose the center of the bridge as the vertex of the parabola and set up a coordinate system with the x-axis representing the distance from the center of the bridge and the y-axis representing the height of the cable.
Since the vertex of the parabola is at the origin (0, 0), we can write the equation of the parabola in vertex form as:
y = a * x^2
Now, let's find the value of a.
Since the towers are 20 feet high, we can find the equation of the parabola by substituting the coordinates of one of the towers.
When x = 150 (75 feet from the center), y = 20 (the height of the tower).
Plugging these values into the equation gives us:
20 = a * 150^2
Simplifying the equation:
20 = a * 22500
a = 20 / 22500
a = 1/1125
So, the equation of the parabola is:
y = (1/1125) * x^2
To find the height of the cable at a point 75 feet from the center, plug x = 75 into the equation:
y = (1/1125) * (75)^2
y = (1/1125) * 5625
y = 5
Therefore, the height of the cable at a point 75 feet from the center of the bridge is 5 feet.
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