Water spraying from a large fountain follows a path modelled by the function h(d) = sa001-1.jpgd2 + 4d, for h ³ 0, where h is the height of the water above the fountain jet and d is the horizontal distance of the water from the fountain jet, both in Water spraying from a large fountain follows a path modelled by the function h(d) = -2/3d^2 + 4d, for h > 0, where h is the height of the water above the fountain jet and d is the horizontal distance of the water from the fountain jet, both in meters.
a) Graph the function (do not need to hand in) and use the graph to determine the maximum height of the fountain. (2 marks).
b)What horizontal distance does the water travel from the fountain jet before it reaches the maximum height? (2 marks)
You must have access to some graphing utility. Here is a good one:
http://www.wolframalpha.com/input/?i=-2%2F3+d^2+%2B+4d&a=*MC.d^2-_*Variable-&a=UnitClash_*d.*Days.dflt--
a) To graph the function h(d) = -2/3d^2 + 4d, we can plot points to create a curve. We can choose various values of d and calculate the corresponding values of h.
Let's calculate h for some values of d:
When d = 0, h = -2/3(0)^2 + 4(0) = 0
When d = 1, h = -2/3(1)^2 + 4(1) = 5 1/3
When d = 2, h = -2/3(2)^2 + 4(2) = 6 2/3
When d = 3, h = -2/3(3)^2 + 4(3) = 6
When d = 4, h = -2/3(4)^2 + 4(4) = 5 1/3
We can continue finding more points, but let's plot these points on a graph:
```
h
^
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|_____________________ d
0 4 8
```
From the graph, we can see that the curve reaches its maximum height around d = 2. Therefore, we can determine the maximum height of the fountain to be approximately h = 6 2/3 meters.
b) To find the horizontal distance at which the water reaches the maximum height, we need to find the value of d when h is at its maximum.
To find the vertex of the parabola, we can use the formula d = -b/2a, where a = -2/3 and b = 4.
d = -4/2(-2/3) = -4/(-4/3) = 3
Therefore, the water travels a horizontal distance of 3 meters from the fountain jet before it reaches the maximum height.
To find the maximum height of the fountain and the horizontal distance traveled before reaching the maximum height, we need to understand the equation given and analyze its graph.
a) To graph the function, we can use any graphing tool or software such as Desmos or Wolfram Alpha. Here's a step-by-step explanation of how to graph the function:
1. Open a graphing tool or software.
2. Choose the coordinate system and set the axes for "h" on the vertical axis and "d" on the horizontal axis.
3. Use the equation h(d) = -2/3d^2 + 4d to create points and plot them on the graph.
4. Connect the points to form a smooth curve.
As explained earlier, we won't be able to create the actual graph as it requires a visual element, but by following the steps above, you should be able to see the graph of the function.
To determine the maximum height of the fountain using the graph:
- The maximum height of the fountain corresponds to the highest point on the graph.
- Locate the vertex (highest point) on the graph. The vertex is the peak of the curve, where the slope changes from positive to negative (or vice versa).
- The y-coordinate of the vertex represents the maximum height of the fountain.
- Read the y-coordinate at the highest point of the graph to find the maximum height.
b) To find the horizontal distance traveled before reaching the maximum height, we need to determine the x-coordinate of the vertex. The x-coordinate of the vertex represents the horizontal distance at which the maximum height occurs.
- Locate the vertex (highest point) on the graph.
- Read the x-coordinate at the highest point of the graph to find the horizontal distance traveled before reaching the maximum height.
By analyzing the graph, you should be able to determine the maximum height of the fountain and the horizontal distance traveled before reaching the maximum height.