Here's my question:

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)

Here's my work on the question:
-x^2+15x=4x-12
x^2-11-12=0
(x-12)(x+1)=0

Can you please help me with this one. help would be much appreciated!

Looks a lot like my work. But we'll let that go...

Surely you can now solve the equation! If two numbers have a product of zero, one or the other must be zero.

So, either

x-12 = 0 --> x=12
x+1 = 0 --> x = -1

Clearly the solution is x=12. That is, the firework will hit on the hill 12 feet horizontally from where it was shot.

If you want to find the height, then plug that value into either function, since that is where they are equal.

To calculate the point where the fireworks will land on the hill, we need to find the intersection of the equation of the slope of the hill and the path of the fireworks.

Given:
Equation of the slope of the hill: y = 4x - 12
Equation of the path of the fireworks: y = -x^2 + 15x

To find the point of intersection, we set the two equations equal to each other and solve for x:

- x^2 + 15x = 4x - 12

Rearrange the equation to get a quadratic equation equal to zero:

x^2 - 11x - 12 = 0

Since the equation is not easily factorable, we can use the quadratic formula to find the values of x:

x = (-b ± √(b^2 - 4ac)) / (2a)

For our quadratic equation, a = 1, b = -11, and c = -12.

Substituting these values into the quadratic formula:

x = (-(-11) ± √((-11)^2 - 4(1)(-12))) / (2(1))
x = (11 ± √(121 + 48)) / 2
x = (11 ± √169) / 2
x = (11 ± 13) / 2

Solving for both values of x:

x₁ = (11 + 13) / 2 = 12
x₂ = (11 - 13) / 2 = -1

So, we have two potential x values where the fireworks could land: x = 12 and x = -1.

To find the corresponding y values, we substitute these x values into either equation:

For x = 12:
y = -x^2 + 15x
y = -(12^2) + 15(12)
y = -144 + 180
y = 36

The point where the fireworks will land on the hill is (12, 36).

Note: We don't need to calculate the y value for x = -1, as that point is not on the hill's slope.