In Gabriel's village there is an electrical wire suspended between two poles across a gully.the poles are on the edge of the gully and the wire is at the same height on both sides of the gully. The curve of the wire can be modeled by the equation h = (d - 6)² + 3
Where h is the height, in meters, of the wire above the floor of the gully and d is the distance, in meters, from the left-hand post.
Use this information to answer the following questions:
A. Draw the graph of h = (d - 6)² + 3 for the values of distance, d, from 0 to 12.
B. What is the minimum height that the wire is above the floor of the gully?
C. a) how far apart are the two poles?
b) how is this shown on the graph?
To draw the graph of the equation h = (d - 6)² + 3, we can create a coordinate system where the horizontal axis represents the distance (d) and the vertical axis represents the height (h).
A. Plotting the graph for d values from 0 to 12:
- For d = 0, we can substitute it into the equation:
h = (0 - 6)² + 3
h = 36 + 3
h = 39
So the first point on the graph is (0, 39).
- For d = 1:
h = (1 - 6)² + 3
h = 25 + 3
h = 28
The second point on the graph is (1, 28).
- Continue in the same way for d = 2, 3, 4, ..., 12 to get the remaining points.
Plot the points and connect them with a smooth curve. The resulting graph should look like a parabola opening upwards.
B. The minimum height that the wire is above the floor of the gully can be found by identifying the vertex of the parabola. In this case, the vertex occurs when d = 6 because of the equation's form (d - 6)².
When d = 6, h = (6 - 6)² + 3 = 3.
So the minimum height that the wire is above the floor of the gully is 3 meters.
C. a) The distance between the two poles can be calculated by finding the x-coordinate values of the points where the parabola intersects the x-axis. In this case, the parabola does not intersect the x-axis, so there is no "zero" value for d. Therefore, the distance between the two poles is undefined.
C. b) On the graph, the distance between the two poles is shown by the x-axis range covered by the parabola, which in this case is from d = 0 to d = 12.
A. To draw the graph of the equation h = (d - 6)² + 3 for the values of distance, d, from 0 to 12, we need to plot the values of h corresponding to each value of d.
Let's substitute different values of d and calculate the corresponding values of h:
For d = 0,
h = (0 - 6)² + 3
h = ( - 6)² + 3
h = 36 + 3
h = 39
So, when d = 0, h = 39.
For d = 1,
h = (1 - 6)² + 3
h = ( - 5)² + 3
h = 25 + 3
h = 28
So, when d = 1, h = 28.
Continuing this process for each value of d from 0 to 12, we get the following table:
d | h
--|--
0 | 39
1 | 28
2 | 19
3 | 12
4 | 7
5 | 4
6 | 3
7 | 4
8 | 7
9 | 12
10| 19
11| 28
12| 39
Now, let's plot these points on the graph:
On the x-axis, we have the values of d ranging from 0 to 12.
On the y-axis, we have the corresponding values of h ranging from 3 to 39.
Connecting these points will give us a curve representing the graph of the equation h = (d - 6)² + 3.
B. To find the minimum height that the wire is above the floor of the gully, we need to find the lowest point on the graph. From the table above, we can see that the lowest point on the graph is when d = 6, which corresponds to h = 3.
Therefore, the minimum height that the wire is above the floor of the gully is 3 meters.
C.
a) To find how far apart the two poles are, we need to find the difference in the values of d at the points where the height is the same on both sides of the gully.
From the table above, we can see that when h = 3, the corresponding values of d are 6 and 6.
Therefore, the two poles are 6 meters apart.
b) On the graph, this is represented by the point where h = 3, which is the lowest point on the curve. The line segment connecting the points where h = 3 on both sides of the graph will show the distance of 6 units between the poles.
A. To draw the graph of the equation h = (d - 6)² + 3, we need to plot various points on a coordinate plane. The x-axis represents the distance (d) and the y-axis represents the height (h).
1. Start by listing the values of d from 0 to 12:
d = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12
2. Substitute these values of d into the equation to find the corresponding values of h. For example:
For d = 0: h = (0 - 6)² + 3 = 9
For d = 1: h = (1 - 6)² + 3 = 16
Repeat this process for each value of d.
3. Plot the coordinates (d, h) on the graph. Connect the points to form the curve.
B. To find the minimum height of the wire above the floor of the gully, we need to determine the lowest point on the graph. In this case, since the equation is in the form h = (d - 6)² + 3, we can observe that the vertex of the parabola represents the minimum point.
1. Convert the equation to the vertex form: h = a(d - h)^2 + k
h = (d - 6)² + 3 (already in the required form)
2. Compare this form with the standard form: h = a(x - h)^2 + k
We can see that the vertex is (h, k) = (6, 3).
3. Hence, the minimum height of the wire above the floor of the gully is 3 meters.
C. a) To find the distance between the two poles, we need to determine the range of values on the x-axis where the curve is defined. In this case, since the x-axis represents the distance (d), the poles are located at the values of d where the curve intersects the x-axis.
1. Set the equation equal to zero: h = (d - 6)² + 3 = 0
2. Solve for d to find the values where the curve intersects the x-axis.
3. In this case, the solutions are complex numbers, indicating that the curve does not intersect the x-axis. Therefore, we can conclude that the two poles are infinitely far apart.
C. b) On the graph, the fact that the two poles are infinitely far apart is represented by the graph not intersecting the x-axis.