Of a parabolic arch and is to have a span of 100 feet. The height of the arch a distance of 40 feet from the center is to be 10 feet. Find the height of the arch at its center.

y = -0.11x^2 + 1.111x

y = 28 ft   Height at center

Step-by-step explanation:

Given:-

- The span of the arc is = 100 ft

- The height of the arch is 40 ft at 10 ft from center.

Find:-

- The equation of parabolic arch and the height of the arch at center.

Solution:-

- We will take the height y as a function of width x of the parabolic arch. The general equation of the arch is such that it passes through origin. The equation is given in the form as:

                               y = f(x) = ax^2 + bx

Where,

           a, b, and c are constants to be determined.

- We will use the condition i.e the span of entire arch is 100 ft. So we could say that y = 0 for x = 100 ft. Then we have:

                               0 = f(100) = a(100)^2 + b(100)   1

- Using second condition i.e y = 10 ft at 40 ft from center. Since, due to symmetry we know that center lies at x = 50 ft. Then y = 10 ft at x = 10 ft. The condition can be expressed in the form:

                               10 = f(10) = a(10)^2 + b(10)   2

- Solving the 2 Equations simultaneously, we have:

                               0 = a(100)^2 + b(100)

                               10*10 = a*10*(10)^2 + b(10)*10

                               100 = a(10)^3 + b(100)

- Subtract both equations:

                               100 = a*(10^3 - 100^2)

                               a = 100 / ( 1000 - 10000)

                               a = -0.11

- Then using a = -0.11 evaluate b:

                               -1.11 + 10b = 10

                                b = 11.11 / 10 = 1.111

- The equation of the parabola is:

                                y = -0.11x^2 + 1.111x                                

-The height of the arch at center where x = 50 ft.

                                y = -0.11(50)^2 + 1.111(50)

                                y = -27.5 + 55.5

                                y = 28 ft                                                        

- The height of the parabolic arch at center is given as y = 28 ft.

452,489 to the nearest thousand is 452,000