The cable in the center portion of a bridge is supported as shown in the figure to form a parabola. The center support is 10 feet high, the tallest supports are 210 feet high, and the distance between the two tallest supports is 400 feet. Find the height of the remaining supports if the supports are evenly spaced. (Ignore the width of the supports.)
With no diagram, I am left to guess that the height of the center parabola is 200.
So I guess that we have the following points on the curve;
(-200,0), (0,200),(200,0)
So, clearly y=400-ax^2
Since y(200)=0, a=1/00
Thus, the height of the support at x=h is 210-y(h)
Well, aren't those some tall supports! I hope they don't develop a fear of heights. Anyway, let's math our way through this.
We know that the cable forms a parabola. A parabola has the general equation y = ax^2 + bx + c. In our case, the vertex of the parabola is at (0, 10) since the center support is 10 feet high. So we can write the equation as y = ax^2 + 10.
Now, we also know that the tallest supports are 210 feet high, which means they are 200 feet above the center support. Since the distance between the two tallest supports is 400 feet, we can conclude that the distance from the center to each tallest support is 200 feet.
To find the equation of the parabola, we need two points on the parabola. Let's use the two tallest supports as our points. The first point is (-200, 210) and the second point is (200, 210). Plugging these values into the equation, we get:
210 = a*(-200)^2 + 10
210 = a*(40000) + 10
200 = a*(40000)
Dividing both sides by 40000, we find a = 0.005.
Now that we have the value of a, we can find the height of the remaining supports. We know that the distance between each support is 400 feet, so let's call the distance from the center support to any support position x. The height of the support can be calculated as:
y = 0.005x^2 + 10
If we plug in any value of x within the range (-200, 200), we'll get the corresponding height of the support. So, for example, if we consider x = 20 (still within the range), we find:
y = 0.005(20)^2 + 10
y = 0.005(400) + 10
y = 2 + 10
y = 12
So, the height of the support at x = 20 feet is 12 feet. You can use the same equation to find the heights of the remaining supports by choosing different values of x.
I hope that helps, and remember, even bridges need a little support and a lot of good jokes!
To determine the height of the remaining supports, we can use the properties of a parabola.
1. Let's define the origin at the center support, with the x-axis extending in both directions.
2. Since the parabola is symmetric, the equation of the parabola can be written as y = ax^2, where a is a constant that needs to be determined.
3. We can find the equation of the parabola using the two given points (0, 10) and (-200, 210).
4. Substituting the coordinates of the first point, we get 10 = a(0^2), which simplifies to a = 10.
5. Now we can find the y-coordinate of any point on the parabola by substituting its x-coordinate into the equation y = 10x^2.
6. The distance between the two tallest supports is 400 feet, so the x-coordinate of one of the tallest supports is 200 feet (-200 + 400/2).
7. Substituting x = 200 into y = 10x^2, we find that the height of one of the tallest supports is 10 * (200^2) = 400,000 feet.
8. Since the supports are evenly spaced, we can find the height of the remaining supports by subtracting the height of one of the tallest supports from the height of the center support.
9. The height of the remaining supports is 400,000 - 10 = 399,990 feet.
Therefore, the height of the remaining supports is 399,990 feet.
To find the height of the remaining supports, we need to first determine the equation of the parabolic shape formed by the cable.
Let's assume that the vertex of the parabola is at the center support, which has a height of 10 feet. The height of the tallest supports, which are equidistant from the center support, is 210 feet. The distance between the two tallest supports is given as 400 feet.
Since the vertex of the parabola is at the center support and the height of the tallest supports is 210 feet, we can use the vertex form of a parabolic equation, which is given by:
y = a(x - h)^2 + k
where (h, k) represents the vertex of the parabola.
In our case, the equation becomes:
y = a(x - 0)^2 + 10
The distance between the two tallest supports is 400 feet. Since the supports are evenly spaced, we can consider half of this distance, 200 feet, as the value of x. Plugging this value into the equation, we get:
210 = a(200 - 0)^2 + 10
Simplifying further:
200^2a = 200
a = 1/400
Now we have the value of a, we can substitute it back into the equation:
y = (1/400)(x - 0)^2 + 10
This equation represents the shape of the cable on the bridge.
To find the height of the remaining supports, we need to calculate the height at regular intervals. The supports are evenly spaced, so we can choose any interval we prefer. Let's choose an interval of 100 feet.
Plugging in different values of x into the equation, we can find the corresponding heights. For example, when x = 100:
y = (1/400)(100 - 0)^2 + 10
= (1/400)(10000) + 10
= 25 + 10
= 35 feet
So, the height of the first remaining support is 35 feet. Similarly, we can find the heights at x = 200, 300, 400, and so on, by repeating the above calculation.
By continuing this process and calculating the heights at regular intervals, we can determine the height of each remaining support on the bridge.