Question

A rocket travels along a curve that is given by the equation y=3x2+7x-27. Will the rocket reach a height of 30 meters? What is the fastest method to check if the rocket can reach the given height and return to Earth?

Answer 1

To check if the rocket will reach a height of 30 meters, we can analyze the given equation and solve for the x-values when y=30 to determine if the rocket reaches the desired height. Calculus and graphing techniques can be used to find critical points and visually evaluate the rocket's trajectory.

Step-by-step explanation:

To check if the rocket will reach a height of 30 meters, we need to find the maximum height the rocket reaches by analyzing the equation y = 3x^2 + 7x - 27.

By comparing the equation with our target height of 30 meters, we can set up the equation:
30 = 3x^2 + 7x - 27

Simplifying and solving this quadratic equation will give us the x-values (or times) when the rocket reaches a height of 30 meters. If there are any valid solutions, it means the rocket can reach the given height.

To find the fastest method to check if the rocket can reach the given height and return to Earth, we can evaluate the rocket's trajectory using calculus or graphing techniques. Calculus allows us to find critical points on the graph, such as maximum height, and determine if the rocket reaches the desired height. Graphing the equation using software or a graphing calculator can visually show us the rocket's trajectory and whether it reaches 30 meters.