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.