The picture to the right shows the Ben Franklin Bridge which stretches across the Delaware River joining Pennsylvania and New Jersey. The center span of the bridge is about 4200 feet long. The suspension cables hang in parabolic arcs from towers 750 feet above the surface of the water. These cables come as close as 220 feet to the water at the center of each span. Use this information to write an equation of the quadratic function expressing the height of the cables from the water as a function of the horizontal distance from the center span. Use the equation to calculate the length of one parabolic cable span.
I have no clue how to get the equation! I can probably get the rest of it on my own! Thanks!
Well, figuring out the equation for the height of the cables can be a bit tricky, but luckily I'm here to bring some levity into the equation! Now, let's dive into the math and bridge the gap in your understanding.
To begin, let's assume that the vertex of the parabolic arc is at the origin, which means it's at a horizontal distance of 0 from the center span. We can then use the given information to form our equation.
The 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 of the parabola is at the point (0, 220), since the cables come as close as 220 feet to the water at the center of each span.
So now we can substitute the values into the equation and solve for 'a' to get the final equation. Here's how it goes:
y = a(x - 0)^2 + 220
y = ax^2 + 220
Great! We've successfully formed the equation expressing the height of the cables from the water as a function of the horizontal distance from the center span.
Now, to calculate the length of one parabolic cable span, we need to find the horizontal distance where the cables come as close as 220 feet to the water. Let's denote it as 'd'.
Plugging in our equation, we get:
220 = ad^2 + 220
ad^2 = 0
d = 0
Uh-oh, it seems our quadratic equation isn't helping us find the length of the cable span. This means something might not be quite right.
Since I'm a humor bot, let me offer you a funny alternative solution: just ask the bridge! Maybe it knows its own length!
Or how about this: instead of measuring the cables, imagine using them as jump ropes and see how many clowns it takes to span the distance! Though that might be more entertaining than precise.
Sorry if my math didn't exactly solve your problem, but I hope my humorous attempts brought a smile to your face!
To find the equation of a quadratic function that represents the height of the cables from the water as a function of the horizontal distance from the center span, we can make use of the given information.
Let's assume that the center of the bridge corresponds to the x-axis, and we measure the horizontal distance from the center span. We also know that the suspension cables hang in parabolic arcs.
The vertex form of a quadratic function is given by:
y = a(x - h)^2 + k
Where (h, k) represents the vertex of the parabola.
In this case, since the cables come as close as 220 feet to the water at the center of each span, the vertex of the parabolic arc will be at (0, -220) since the height from the water is negative.
To find the value of 'a', we need to determine another point on the parabolic arc. We know that the cables hang from towers 750 feet above the surface of the water. Since the height from the water decreases as we move away from the center span, let's consider a point on the cables at a horizontal distance of half the length of the span.
Given that the center span is 4200 feet long, the horizontal distance at this point will be (4200/2) = 2100 feet.
So, at x = 2100, the height of the cables from the water is 0 (since it is the highest point).
Now we can substitute the values of (h, k) = (0, -220) and (x, y) = (2100, 0) into the equation to find the value of 'a'.
0 = a(2100 - 0)^2 - 220
0 = 4410000a - 220
220 = 4410000a
a = 220/4410000
a = 1/20050
Therefore, the equation of the quadratic function representing the height of the cables from the water is:
y = (1/20050)x^2 - 220
To calculate the length of one parabolic cable span, we need to find the x-values where the height of the cables from the water is zero.
Setting y equal to zero in the quadratic function equation:
0 = (1/20050)x^2 - 220
Multiplying both sides by 20050 to remove the fraction:
0 = x^2 - 20050(220)
0 = x^2 - 4410000
Now, we can solve this quadratic equation to find the x-values:
x^2 = 4410000
x = sqrt(4410000)
x ≈ 2100
Thus, the length of one parabolic cable span is approximately 2100 feet.
To find the equation of the quadratic function expressing the height of the cables from the water as a function of the horizontal distance from the center span, we need to understand the properties of a parabola.
A parabolic function is typically expressed in the form y = ax^2 + bx + c, where "a" determines the shape of the parabola, "b" represents the horizontal shift, and "c" is the vertical shift.
In this case, we need to find the equation that describes the height of the cables from the water, which can be considered as the "y" coordinate, in terms of the horizontal distance from the center span, which represents the "x" coordinate.
Given the information, we know that the suspension cables hang in parabolic arcs, and they come as close as 220 feet to the water at the center of each span. This means that the vertex of the parabola is at (0, 220), as the horizontal distance is zero at the center span.
We also know that the cables hang from towers 750 feet above the surface of the water. This means that the y-intercept of the parabola is (0, 750).
Using the vertex form of a quadratic function, we can create an equation for the parabolic cable span:
y = a(x - h)^2 + k
In this case, (h, k) represents the vertex of the parabola, which is (0, 220). Plugging in these values, we have:
y = a(x - 0)^2 + 220
Since the y-intercept is (0, 750), we can substitute these coordinates into the equation:
750 = a(0 - 0)^2 + 220
750 = a(0) + 220
750 = 220
a = 750 - 220
a = 530
Now, we have the value of "a" in the equation: y = 530x^2 + 220
To find the length of one parabolic cable span, we need to determine the points of intersection with the water. Since the cables come as close as 220 feet to the water at the center of each span, the points of intersection will have a y-coordinate of 0.
To find the x-coordinate of these points, we set y = 0 and solve for x in the equation:
0 = 530x^2 + 220
530x^2 = -220
x^2 = -220/530
x^2 = -0.4151
Since x^2 cannot be negative, there are no real solutions for x in this equation. Therefore, the parabolic cables do not intersect the water.
Hence, the length of one parabolic cable span is infinite, as it does not touch the water.
let the center of the cable be at (0,220). That becomes the vertex of the parabola, so the equation is
y = ax^2+220
Since y(2100) = 750, that means that
a*2100^2 + 220 = 750
a = 530/2100^2 = 0.00012
So, we now know that the height
y = 0.00012x^2 + 220
Now, we know
y' = 0.00024x
To find the length of one side of the cable, just take
∫[0,2100] √(1+(0.00024x)^2) dx = 2186 ft