SOLUTION: A bridge is built in the shape of a parabolic arch and is to have a span of 100 feet. The height of the arch at a distance of 40 feet from the center is to be 10 feet. Assume that

Algebra ->  Quadratic-relations-and-conic-sections -> SOLUTION: A bridge is built in the shape of a parabolic arch and is to have a span of 100 feet. The height of the arch at a distance of 40 feet from the center is to be 10 feet. Assume that       Log On


   

Question 1185067: A bridge is built in the shape of a parabolic arch and is to have a span of 100 feet. The height of the arch at a distance of 40 feet from the center is to be 10 feet. Assume that the ground is the x-axis and the y-axis as the axis of the arch.
a. How high is the arch at its center approximated in two decimal places?
b. What is the horizontal length approximated to two decimal places of the arch from its axis when it’s 15 feet high?
Answer by greenestamps(13200) About Me  (Show Source):
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With the origin at ground level at the center of the arch, the equation of the parabola is of the form

y=ax%5E2%2Bb

We know that the height is 0 50 feet from the center of the arch and 10 feet 40 feet from the center. Use those two points to determine the coefficients a and b.

0=2500a%2Bb
10=1600a%2Bb

-10=900a
a=-10%2F900=-1%2F90

0=2500%28-1%2F90%29%2Bb
b=2500%2F90=250%2F9

The equation of the parabola is

y=%28-1%2F90%29x%5E2%2B250%2F9

a. The height at the center of the arch is the y value when x=0.
y=250%2F9

b. To find the width of the arch when the height is 15, find the x value when y is 15, then double that answer:

15=%28-1%2F90%29x%5E2%2B250%2F9
...

Numerical calculations or a graphing calculator should give you x=33.91 (to 2 decimal places), so the width of the arch when the height is 15 feet is 67.82 feet.