Fireworks are to be launched from a platform at the base of a hill. Using the top of the platform as the origin and taking some measurements, it was determined that the cross-section of the slope of one side of the hill is y=4x-12 If the path of the fireworks is y=-x^2+15x, calculate the point where the fireworks will land on the hill. (Show your work - 3 marks)
To find the point where the fireworks will land on the hill, we need to find the intersection point between the path of the fireworks and the slope of the hill.
First, let's set the two equations equal to each other and solve for x:
4x - 12 = -x^2 + 15x
Rearranging the equation, we have:
x^2 - 11x + 12 = 0
Now we can factor the quadratic equation:
(x - 3)(x - 4) = 0
This gives us two possible values for x: x = 3 and x = 4.
To determine which value of x corresponds to the point where the fireworks will land on the hill, we can substitute each value of x back into one of the original equations and solve for y.
If we substitute x = 3 in the equation y = -x^2 + 15x:
y = -(3)^2 + 15(3) = 6
So, when x = 3, y = 6, which means one possible landing point is (3, 6).
If we substitute x = 4 in the equation y = -x^2 + 15x:
y = -(4)^2 + 15(4) = 4
So, when x = 4, y = 4, which means another possible landing point is (4, 4).
Therefore, the fireworks will land on the hill at the points (3, 6) and (4, 4).