Water spraying from a large fountain follows a path modeled 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 metres.
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)
To graph the function h(d) = -2/3d^2 + 4d, we can first plot a few points and then connect them with a smooth curve.
Let's choose a few values for d and calculate the corresponding values of h(d):
For d = 0, h(0) = -2/3(0)^2 + 4(0) = 0
For d = 1, h(1) = -2/3(1)^2 + 4(1) = 1 1/3
For d = 2, h(2) = -2/3(2)^2 + 4(2) = 2 2/3
For d = 3, h(3) = -2/3(3)^2 + 4(3) = 3 1/3
For d = 4, h(4) = -2/3(4)^2 + 4(4) = 4
Now, plot these points on a graph where the horizontal axis represents d and the vertical axis represents h(d). Connect the points with a smooth curve:
- The point (0, 0)
- The point (1, 1 1/3)
- The point (2, 2 2/3)
- The point (3, 3 1/3)
- The point (4, 4)
The resulting graph should be a parabolic shape opening downwards.
To determine the maximum height of the fountain, we need to find the vertex of the parabola. The x-coordinate of the vertex can be found using the formula x = -b/2a, where a and b are the coefficients in the equation h(d) = -2/3d^2 + 4d.
In our case, a = -2/3 and b = 4. Substituting these values into the formula, we get:
x = -(4) / 2(-2/3) = -4 / (-4/3) = 3
So, the horizontal distance at which the water reaches its maximum height is 3 meters.
Now, substitute this value back into the equation h(d) = -2/3d^2 + 4d to find the maximum height:
h(3) = -2/3(3)^2 + 4(3) = -2/3(9) + 12 = -6 + 12 = 6
Hence, the maximum height of the fountain is 6 meters.