A bridge is built in the shape of a parabolic arch. The bridge has a span of s= 160 feet and a maximum height of h= 25 feet. Choose a suitable rectangular coordinate system and find the height of the arch at distances of 10, 30, and 50 feet from the center.
10=
30=
50=
If it has a span of 160', then the equation of the parabola is
f(x)=ax(160-x)
where a is a constant, and x=0 and x=160 are the zeroes which correspond to points where the arch reaches ground/water.
The constant "a" can be found by the fact that f(80)=25.
Post your answer for a check if you wish.
Yes, the answers are all correct.
You have also correctly taken the origin at the centre of the span, which is logical since the question requires the height at distances from the centre.
Check that at negative distances the heights are the same.
To find the height of the arch at distances of 10, 30, and 50 feet from the center of the bridge, we can use the equation of a parabolic arch in vertex form, which is
y = a(x-h)^2 + k
In this equation, (h, k) represents the vertex or highest point of the parabola, and 'a' determines the steepness of the arch. In this case, the vertex is at (0, 25) because the maximum height of the bridge is 25 feet.
To find 'a', we can use the fact that the bridge has a span of 160 feet, which means that the arch touches the ground at (±80, 0) points. Substituting these coordinates into the equation, we can find 'a'.
0 = a(±80-0)^2 + 25
0 = a(80^2) + 25
0 = 6400a + 25
Solving this equation, we find:
6400a = -25
a ≈ -0.0039
Now we can use this value of 'a' to find the height of the arch at distances of 10, 30, and 50 feet from the center.
1. For x = 10:
y = (-0.0039)(10-0)^2 + 25
y = (-0.0039)(100) + 25
y = -0.39 + 25
y ≈ 24.61 feet
Therefore, the height of the arch at a distance of 10 feet from the center is approximately 24.61 feet.
2. For x = 30:
y = (-0.0039)(30-0)^2 + 25
y = (-0.0039)(900) + 25
y = -3.51 + 25
y ≈ 21.49 feet
Therefore, the height of the arch at a distance of 30 feet from the center is approximately 21.49 feet.
3. For x = 50:
y = (-0.0039)(50-0)^2 + 25
y = (-0.0039)(2500) + 25
y = -9.75 + 25
y ≈ 15.25 feet
Therefore, the height of the arch at a distance of 50 feet from the center is approximately 15.25 feet.
10 feet= 24.61 ft
30 feet= 21.48 ft
50 feet= 15.23 ft