Solve Applied Problems Involving Parabolas
-An experimental model for a suspension bridge is built. In one section, cable runs from the top of one tower down to the roadway, just touching it there, and up again to the top of a second tower. The towers are both 6.25 inches tall and stand 50 inches apart. Find the vertical distance from the roadway to the cable at a point on the road 7.5 inches from the lowest point of the cable.
A) 0.56 in.
B) 2.25 in.
C) 0.76 in.
D) 0.36 in.

Question

Quizplus · Accepted Answer

The equation of a parabola in vertex form is $y = a(x-h)^2 + k$, where $(h, k)$ is the vertex of the parabola. Here, the lowest point of the cable, which is the vertex of the parabola, is at the midpoint between the two towers, and the roadway is the x-axis. The towers are 50 inches apart, so the vertex is at $x = 25$ inches (halfway between the towers), and we can assume $y = 0$ at the vertex since the cable just touches the roadway there. The towers, being 6.25 inches tall, give us another point on the parabola: $(0, 6.25)$ and $(50, 6.25)$.To find the coefficient $a$, we use one of these points. Let's use $(0, 6.25)$:$$6.25 = a(0-25)^2 + 0$$$$6.25 = 625a$$$$a = \frac{6.25}{625} = 0.01$$So, the equation of the parabola is $y = 0.01(x-25)^2$.To find the vertical distance from the roadway to the cable at a point 7.5 inches from the lowest point of the cable, we substitute $x = 25 + 7.5 = 32.5$ into the equation:$$y = 0.01(32.5-25)^2$$$$y = 0.01(7.5)^2$$$$y = 0.01(56.25)$$$$y = 0.5625$$Therefore, the vertical distance is approximately 0.56 inches.