2 towers of a suspension bridge are 1000 meters apart and are 160 meters high. The cable between 2 towers hangs in a shape of parabola, which at its lowest is 40meters above the road.
A. Find the equation of the parabola, if we put a coordinate system so that the axis of the parabola is the y-axis and the road is on the x axis.
B. How high is the cable 100 meter from a tower?
clearly, the vertex of the parabola is at (0,40). That means the equation is
y = ax^2 + 40
So now all you have to do is find a. You have the point (500,160) so just plug that in and you can get a.
Once you have a, then you just need y(400).
A. To find the equation of the parabola, let's define the vertex of the parabola as the point (0, h), where h is the height of the cable at its lowest point (40 meters above the road).
Since the parabola is symmetric with respect to the y-axis, its equation is of the form y = a*x^2 + h, where a is a constant. To find the value of a, we need to find another point on the parabola.
Let's consider the point (500, 0) which is halfway between the two towers. At this point, the cable is at road level (0 meters above the road). Plugging these values into the equation, we get:
0 = a*(500^2) + h
0 = 250000a + h
Since h = 40 (given), we can substitute it in:
0 = 250000a + 40
Solving for a, we get:
250000a = -40
a = -40/250000
a = -0.00016
Now we have the equation of the parabola:
y = -0.00016*x^2 + 40
B. To find the height of the cable 100 meters from a tower, we need to find the corresponding y-coordinate on the parabola.
Let's consider the point (100, y). Plugging this point into the equation, we get:
y = -0.00016*(100^2) + 40
y = -0.00016*10000 + 40
y = -1.6 + 40
y = 38.4
Therefore, the cable is 38.4 meters high 100 meters away from a tower.
A. To find the equation of the parabola, let's assume the vertex of the parabola is at the origin (0,0). Since the cable hangs in a shape of a parabola, its equation can be represented as y = ax^2, where a is a constant.
We know that at the vertex (0,0), the y-coordinate is 40 meters above the road. So, we have the point (0,40).
To further determine the value of a, we need another point on the curve. We can use the point (500,160), which is halfway between the two towers. Since the two towers are 1000 meters apart and the cable hangs symmetrically, the x-coordinate of this point is half of the distance between the two towers.
Substituting the coordinates of this point into the equation, we have:
160 = a(500)^2
160 = 250,000a
Solving for a:
a = 160/250,000
a = 0.00064
Therefore, the equation of the parabola is y = 0.00064x^2 + 40.
B. To find how high the cable is 100 meters from a tower, we can substitute x = 100 into the equation y = 0.00064x^2 + 40 and solve for y.
y = 0.00064(100)^2 + 40
y = 0.00064(10,000) + 40
y = 6.4 + 40
y = 46.4
Therefore, the cable is 46.4 meters high 100 meters from a tower.
To find the equation of the parabola, we can use the vertex form of a parabola equation: y = a(x-h)^2 + k, where (h, k) represents the vertex.
A. To find the vertex of the parabola, we need to determine the coordinates of the lowest point or the "bottom" of the parabola. We know that the bottom of the parabola is located 40 meters above the road, which means the y-coordinate is 40.
The bridge towers are 1000 meters apart, so the parabola is symmetric. The vertex of the parabola will be at the midpoint between the two towers, which would be (500, 40) since we have the origin set up with the y-axis as the axis of the parabola.
Next, we need to find the value of 'a', which determines the shape and direction of the parabola. Since the parabola opens upwards and the vertex is located below the x-axis (in the negative y-values), 'a' must be positive.
Now, we have the vertex (500, 40) and the form of the equation is y = a(x-500)^2 + 40.
To find the value of 'a', we can substitute the coordinates of one of the towers, let's say (0, 160), into the equation since the tower lies on the parabola:
160 = a(0-500)^2 + 40
160 = 250,000a + 40
250,000a = 160 - 40
a = 120/250,000
a = 0.00048 (approximately)
So, the equation of the parabola is y = 0.00048(x-500)^2 + 40.
B. To find the height of the cable at a distance of 100 meters from the tower, we can substitute x = 100 into the equation and solve for y:
y = 0.00048(100-500)^2 + 40
y = 0.00048(-400)^2 + 40
y = 0.00048(160,000) + 40
y = 76.8 + 40
y = 116.8
Therefore, the height of the cable 100 meters from a tower is approximately 116.8 meters.