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)
same thing again, plug in the straight line equation, and solve
To find the point where the fireworks will land on the hill, we need to find the intersection point of the slope of the hill and the path of the fireworks.
1. Let's set the equations of the slope of the hill and the path of the fireworks equal to each other:
4x - 12 = -x^2 + 15x
2. Rearrange the equation to get a quadratic equation in standard form:
x^2 - 11x + 12 = 0
3. Solve the equation by factoring or using the quadratic formula. In this case, the equation can be factored as:
(x - 3)(x - 4) = 0
Setting each factor equal to zero gives us two possible values for x:
x - 3 = 0 --> x = 3
x - 4 = 0 --> x = 4
4. Now that we have the x-values, let's substitute them back into the equation of the slope of the hill to find the corresponding y-values.
For x = 3:
y = 4(3) - 12 = 0
For x = 4:
y = 4(4) - 12 = 4
5. So, the point where the fireworks will land on the hill is (3, 0) or (4, 4) depending on which side of the hill the fireworks are launched from.