Answer:
1) The distance between the ends of the arch is approximately 98.6
2) The eight of the arc is approximately 12.3
Explanation:
1) The function for the height of the arch, h(x) = -0.005061·x² + 0.499015·x
Where;
x = The distance from the ends of the arch = 0, which gives;
0 = -0.005061·x² + 0.499015·x
Factorizing the above equation, we get;
0 = x·(-0.005061·x + 0.499015)
Which gives;
x = 0 or (-0.005061·x + 0.499015) = 0
-0.005061·x + 0.499015 = 0 gives;
-0.005061·x = -0.499015
x = -0.499015/(-0.005061) ≈ 98.6
Therefore, the height of the arch is zero at distance x = 0 and x = 98.6
Which gives the distance between the ends of the arch = 98.6
2) The height of the arc function h(x) = -0.005061·x² + 0.499015·x, whereby the coefficient of x² is negative, shows that it is ∩-shaped, the coordinates height and therefore, the height, is given by equating the derivative of the function to zero as follows;
d(h(x))/dt = d(-0.005061·x² + 0.499015·x)/dt = 2×(-0.005061)×x + 0.499015
d(h(x))/dt = 0 gives;
2×(-0.005061)×x + 0.499015 = 0
x = -0.499015/(2×(-0.005061)) ≈ 49.3
Therefore, the height of the arc, is the height at the point where x = 49.3
Therefore, we find the height of the arc from the height equation as follows;
h(x) = -0.005061×(49.3)² + 0.499015×49.3 ≈ 12.3
The eight of the arc is approximately 12.3.