The cable in the center portion of a bridge is supported by five pillars to form a parabola. The center support is 10 ft high, the tallest supports are 210 feet high, and the distance between the two tallest supports is 400 ft. Find the height of the two supports equally spaced between the center support and the tallest supports.

Please help and show work :) Thank you in advance!

well, do right 200 foot half of it from the 10 ft level up and add 10 feet at the end

(0,0) , (100,y middle right) , (200,200)

y = k x^2
200 = k (200)^2
k = 1/200

at x = 100
y = (1/200)(100)^2
= (1/2)(100)
= 50
add that ten feet to compensate for moving the origin up at the beginning
height at 100 feet right of center = 60 feet

Well, well, well, we've got ourselves a math problem with some fancy pillars and a bridge! Let's see if we can "support" you with the answer, shall we?

First, let's set up the problem. We have a parabolic cable supported by five pillars, with the center support being 10 ft high and the two tallest supports measuring a whopping 210 ft high. The distance between these two tall pillars is 400 ft.

To find the height of the two supports equally spaced between the center and the tallest supports, we need to figure out the interval between these supports. In other words, we need to divide the distance between the tallest supports (400 ft) by the number of intervals (4). This will give us the space between each support.

So, 400 ft divided by 4 gives us 100 ft. Bingo!

Now, let's use this interval to find the respective heights of the two supports.

Starting from the center support, the height gradually increases until we reach the first equally spaced support. Since there are two equally spaced supports between the center and the tallest supports, we divide the interval distance (100 ft) by 3 to determine the incremental increase in height between each consecutive support.

100 ft divided by 3 is approximately 33.33 ft. Fancy decimals, huh?

Now, let's calculate the height of the first equally spaced support:

10 ft (height of the center support) + 33.33 ft = 43.33 ft

So, the height of the first equally spaced support is approximately 43.33 ft.

Moving on to the second equally spaced support:

43.33 ft + 33.33 ft = 76.66 ft

Hence, the height of the second equally spaced support is approximately 76.66 ft.

And there you have it! The two supports equally spaced between the center support and the tallest supports have heights of approximately 43.33 ft and 76.66 ft, respectively.

I hope this explanation "bridges" the gap of your understanding! If you have any more math conundrums, feel free to ask!

To find the height of the two supports equally spaced between the center support and the tallest supports, we can use the concept of a parabola and the vertex form of its equation.

1. Let's assume the distance from the center support to one of the equally spaced supports is x (in feet).
2. Since there are five supports in total, the distance between the two tallest supports is 400 ft, which means the distance from the center support to one of the tallest supports is 200 ft.
3. Let's represent the height of the tallest supports as h and the height of the equally spaced supports as y. So, the height of the center support (the vertex of the parabola) is 10 ft.
4. The vertex form of a parabola equation is given by: y = a(x - h)^2 + k, where (h, k) is the vertex point of the parabola.
5. In our case, the vertex point is (200, 10), and we need to find the a value.
6. Substituting the vertex point into the equation, we get: 10 = a(200 - 200)^2 + 10
Simplifying, we have: 10 = a * 0 + 10.
This gives us: 10 = 10.
7. Since a * 0 = 0, the value of a does not affect the equation. Therefore, we can consider a as 1. So, the equation becomes: y = (x - 200)^2 + 10.
8. Now, we can substitute x with 100 and 300 to find the heights of the equally spaced supports.
- For x = 100, the equation becomes: y = (100 - 200)^2 + 10 = (-100)^2 + 10 = 10000 + 10 = 10010 ft.
- For x = 300, the equation becomes: y = (300 - 200)^2 + 10 = (100)^2 + 10 = 10000 + 10 = 10010 ft.

Therefore, the height of the two supports equally spaced between the center support and the tallest supports is 10,010 ft.

To find the height of the two supports equally spaced between the center support and the tallest supports, we will use the properties of a parabola.

First, let's label the points:
- A represents the center support point at a height of 10 ft.
- B and C represent the two equally spaced supports we want to find at an unknown height.
- D and E represent the tallest supports at a height of 210 ft.

We are given that the distance between the two tallest supports (BE) is 400 ft. Since B and E are equally spaced from the center support, we know that the distance between the center support and each of the tallest supports (AB or AE) is half of 400 ft, which is 200 ft.

Now, let's use the fact that the shape formed by the cable is a parabola. In a symmetric parabolic shape, the height at the center (A) is the maximum height, and the heights on both sides decrease at an equal rate.

Using this property, we can see that the distance between the tallest supports (BD or DE) is also divided equally. So, BD or DE is 200 ft divided by 2, which is 100 ft.

Now that we have the distances, let's find the heights of the supports B and C.

To find the height of support B, we can subtract the distance BD (100 ft) from the height of the tallest supports (210 ft):
Height of support B = Height of tallest supports - Distance BD = 210 ft - 100 ft = 110 ft.

Similarly, to find the height of support C, we add the distance DE (100 ft) to the height of the tallest supports (210 ft):
Height of support C = Height of tallest supports + Distance DE = 210 ft + 100 ft = 310 ft.

Therefore, the height of the two supports equally spaced between the center support and the tallest supports is 110 ft and 310 ft respectively.