Answer:
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.
