We're given the parametric equations x = 3t^2 and y = -4t + 16 that describe the path of a rocket. We need to find the time it takes for the rocket to hit the ground, which means we need to find the time t when its y-coordinate becomes zero.
Here's how we can do that:
1. Set y to zero:
The rocket hits the ground when its y-coordinate reaches zero, so we need to set the y equation to zero:
-4t + 16 = 0
2. Solve for t:
Now we need to solve this equation for the value of t. We can do this by isolating t:
-4t = -16
t = 16 / -4
t = -4
3. Check for physical feasibility:
The negative value of t doesn't make sense in this context as time cannot be negative. Therefore, we discard this solution.
4. Interpretation:
Since the equation has only one solution for t, this means the rocket only hits the ground once. The time it takes for the rocket to hit the ground is t = 4 units.
Visualizing the Solution:
We can visualize the solution by plotting the parametric equations. The x-axis represents time (t), and the y-axis represents the rocket's height. The point where the curve intersects the x-axis corresponds to the time when the rocket hits the ground.
Image of graph with xaxis labeled time and yaxis labeled height. The graph shows a parabola that intersects the xaxis at one point.Opens in a new window
graph with xaxis labeled time and yaxis labeled height. The graph shows a parabola that intersects the xaxis at one point.
In this specific case, the parabola intersects the x-axis at t = 4, confirming our solution.
The units of time (t) will depend on the units used for the parametric equations.
This approach can be applied to solve similar problems involving parametric equations and finding the time it takes for an object to reach a specific point.
I hope this explanation is more detailed and clarifies any remaining questions you might have!