rocket is fired off the ground at a target 200 metres away. It is designed to reach a height of 250

Question

metres as it heads toward its target on a parabolic path. (6 marks)
a. Find the equation that represents the height off the ground versus the distance travelled for this
rocket.
b. An 85-metre-high building stands between the launch site and the target. The building is only
20 metres from the target. Will the rocket hit or miss the building? Explain or justify your
answer.

Answer 1

a. To find the equation that represents the height off the ground versus the distance traveled for this rocket, we can use the standard form equation of a parabola, which is:
y = a(x - h)² + k

Where (h, k) represents the vertex of the parabola and "a" determines how steep or flat the graph of the parabola is.

In this case, the rocket is fired off the ground at a target 200 meters away and is designed to reach a height of 250 meters. Since the rocket is launched at the ground, the initial height is 0 meters.

Let's choose the vertex to be at the starting point of the rocket, (0, 0). The x-coordinate represents the distance traveled, so the equation becomes:
y = a(x - 0)² + 0
y = ax²

We know that when the rocket is at a distance of 200 meters, its height is 250 meters. Using this information, we can plug in these values to solve for "a":
250 = a(200)²
250 = a(40000)
a = 250 / 40000
a = 0.00625

Therefore, the equation that represents the height off the ground versus the distance traveled for this rocket is:
y = 0.00625x²

b. To determine if the rocket will hit or miss the building, we need to find out if the height of the rocket at a distance of 20 meters is higher or lower than the height of the building (85 meters).

Using the equation we found in part a, we can substitute x = 20 into the equation to find the height of the rocket:
y = 0.00625(20)²
y = 0.00625(400)
y = 2.5

The height of the rocket at a distance of 20 meters is 2.5 meters. Since this height is lower than the height of the building (85 meters), the rocket will miss the building.