Part A.
We know that the green hose can reach up to 8 feet. Now, we have to find the maximum height reached by the water using the red hose. We use the concept of calculus. The maxima or minima of any given equation can be determined by setting the first derivative to zero.
y = -(x - 3)² + 7
y = -(x²-6x+9)+7
y = -x²+6x-2
dy/dx = -2x + 6 = -
x = 3
Substituting x=3 to the original equation,
y = -(3-3)² + 7
y = 7 feet
The red hose can only reach up to 7 feet. Therefore, the green hose can throw the water higher.
Part B. The graph for the green hose is shown in the attached picture. The plants and the hose are 4 ft above the ground. Therefore, the water's direction also starts at 4 feet. Then, the maximum height is 8 feet. The y-coordinate would then be 8+4 = 12 feet. Now, we use these points to determine the equation of the parabola with a general form of
(x - h)² = -4a(y-k)
From the graph, the vertex is found at the maximum height: (5,12). Then, using the point at (10,4):
(10-5)² = -4a(4-12)
4a = -3.125
Therefore, the equation of the parabola is:
(x-5)² = -3..125(y-12)
Part C. The domain is the coverage of all x-value covered by the parabola. So, that starts from 0 to 10. The domain is then [0,10]. The range is the coverage of all the y-values. That means the range is [4,12].
