Harry Potter is caught in a fierce duel between him and his rival Draco Malfoy. Draco just missed Harry with the Expelliarmus spell and Harry fires back with a Flipendus jinx, attempting to send Draco flying. Harry is standind 12 feet from Draco and shot the spell at an angle of 38 degrees from his wand, which is 4 feet above the ground. The Flipendo jinx travels at a rate of 22 ft/sec. Disarmed, as soon as Harry unleashes the spell, Draco starts to run in the opposite direction at a rate of 6 ft/sec. Will the spell die out before it hits Draco? If not at what value of x will the spell hit Draco? Create 2 sets of parametric equations to model both the spell and Draco running.

To determine if the spell will hit Draco and at what value of x, we need to consider the motion of both Harry's spell and Draco running.

First, let's set up the parametric equations for the motion of Harry's spell. We can use the equation of motion in projectile motion:

For horizontal motion:
x_spell(t) = 22t

For vertical motion:
y_spell(t) = 4 + 22t * tan(38°) - 16t^2

Next, let's set up the parametric equations for Draco's running motion:

For horizontal motion:
x_draco(t) = -6t + 12

(Note: Since Draco is running in the opposite direction, we use a negative sign for his velocity.)

Now, let's equate the x-coordinates of Harry's spell and Draco to find when they intersect:

x_spell(t) = x_draco(t)
22t = -6t + 12

Solving the equation, we can find:
28t = 12
t ≈ 0.4286 seconds

So, at approximately 0.4286 seconds, the spell will hit Draco.

To find the value of x at which the spell hits Draco, we substitute this time value into either of the parametric equations. Let's use the equation for Draco's motion:

x_draco(0.4286) = -6(0.4286) + 12
x ≈ 9.14285

Therefore, the spell will hit Draco at a value of x ≈ 9.14285.

Now we have the following parametric equations for the motion:

For the spell:
x_spell(t) = 22t
y_spell(t) = 4 + 22t * tan(38°) - 16t^2

For Draco:
x_draco(t) = -6t + 12

These equations describe the path of the spell and Draco's motion.

To determine whether the Flipendo jinx will hit Draco and at what value of x it will do so, let's break down the problem into two sets of parametric equations: one for the spell and one for Draco running.

First, let's start with the parametric equations for the spell:

x_spell(t) = 4t (where t represents time in seconds)
y_spell(t) = 4 + (22t * sin(38°))
z_spell(t) = 22t * cos(38°)

In these equations, x_spell(t) represents the horizontal distance of the spell from its starting point (Harry's wand), y_spell(t) represents the vertical distance, and z_spell(t) represents the distance from the ground.

Now, let's move on to the parametric equations for Draco running:

x_dracorun(t) = 12 - 6t (where t represents time in seconds)
y_dracorun(t) = 0 (since Draco runs horizontally)
z_dracorun(t) = 0 (since Draco runs on the ground)

In these equations, x_dracorun(t) represents Draco's horizontal position, y_dracorun(t) represents the vertical position (which is constant since Draco runs horizontally), and z_dracorun(t) represents the height from the ground (which is also constant since Draco runs on the ground).

To determine whether the spell hits Draco, we need to find the values of t at which the spell's position intersects with Draco's position. We can do this by setting the x-coordinate and y-coordinate of the spell equal to Draco's x and y at the same time:

4t = 12 - 6t [Equating x-coordinate]
4 + (22t * sin(38°)) = 0 [Equating y-coordinate]

Now, let's solve the equations to find the values of t when the spell hits Draco.

4t = 12 - 6t
10t = 12
t = 12/10
t = 1.2 seconds

Plugging this t-value into the y-coordinate equation, we have:

4 + (22 * 1.2 * sin(38°)) = 0

Simplifying this equation will give us the y-coordinate at the moment of impact.

For the second part of the question, it asks for Draco's x-coordinate at the moment of impact. Using the t-value we obtained earlier:

x_dracorun(1.2) = 12 - 6 * 1.2

Simplifying this equation will give us Draco's x-coordinate at the moment of impact.

By solving these equations, you will be able to determine if the spell will hit Draco and at what value of x it will do so.