a suspenion bridge is in a shape of a parabola and has a length of 3000 feet between the uprights. the roadbed is 50 feet above the river and the lowest point of the curved cable is 25 feet above the road. if the height is 450 feet, how long is the vertical support cable that is located halfway between an upright and the center of the bridge?
y=a(x-1500)^2+b is the form of the equation.
you need to solve for a, b. Well, you know immediately that at x=1500, b has to be 25
Now, at x=0, you know y=450. Solve for a.
Now, with a,b solve for y when x=750
What is the calculus in this?
a parabolic bridge over a road is 4 meter tall 4 meter wide and 5 meter long determain the maximum height of a 2 meter truck can drive under the bridge
To find the length of the vertical support cable located halfway between an upright and the center of the bridge, we can use the properties of a parabola and some basic algebra.
Let's denote the distance from one of the uprights to the center of the bridge as 'a'. Since the length of the bridge between the uprights is given as 3000 feet, we have:
Length of the bridge, L = 2a = 3000 feet
This yields the value of 'a' as follows:
a = 3000 feet / 2 = 1500 feet
Now, let's denote the height of the parabolic shape (measured from the lowest point of the curved cable to the roadbed) as 'h'. Given that the roadbed is 50 feet above the river and the lowest point of the curved cable is 25 feet above the road, we can determine the height 'h':
h = 50 feet + 25 feet = 75 feet
Furthermore, we are provided with the maximum height of the parabolic shape, which is 450 feet.
Next, let's determine the equation of the parabola that represents the shape of the bridge:
The general equation of a parabola in vertex form is:
y = a(x - h)^2 + k
where (h, k) represents the vertex of the parabola. In this case, the vertex is at the maximum height of the parabolic shape, so h is 0 (since the parabola is symmetric about the y-axis) and k is the maximum height, which is 450 feet.
Therefore, the equation of the parabola is:
y = a(x - 0)^2 + 450
= ax^2 + 450
Now, let's find the value of 'a' by using one of the known points on the parabolic curve. The point halfway between an upright and the center of the bridge has a height of h/2 = 75/2 = 37.5 feet.
Substituting this point's coordinates into the equation, we get:
37.5 = a(1500)^2 + 450
Now we can solve for 'a' using algebra:
37.5 = 2250000a + 450
2250000a = 37.5 - 450
2250000a = -412.5
a = -412.5 / 2250000
a ≈ -0.00018333
So, the equation of the parabolic shape of the bridge is:
y = -0.00018333x^2 + 450
Now, to find the length of the vertical support cable located halfway between an upright and the center of the bridge, we need to find the corresponding x-coordinate on the bridge.
To find that, we can equate y to the height of the cable above the roadbed and solve for x:
-0.00018333x^2 + 450 = 37.5
Subtracting 37.5 from both sides:
-0.00018333x^2 = 37.5 - 450
-0.00018333x^2 = -412.5
Dividing both sides by -0.00018333:
x^2 ≈ -412.5 / -0.00018333
x^2 ≈ 2250000000
x ≈ √2250000000
x ≈ 47452.772
So, the horizontal distance from the center of the bridge to the cable is approximately 47452.772 feet.
Finally, since the vertical support cable is located halfway between the center of the bridge and an upright, the total length of the cable can be calculated by doubling the horizontal distance:
Length of the vertical support cable = 2 * 47452.772 feet
= 94891.544 feet (approx.)