Question

NASA launches a rocket at t = 0 seconds. Its height function is given by H(t) = -4.9t^2 + 268t + 325.

At what time does splashdown occur?
How high above sea-level does the rocket get at its peak?
The rocket peaks at how many meters above sea-level?

Answer 1

Splashdown occurs when the height function of the rocket equals zero; this can be found using the quadratic formula. The peak height is determined by finding the vertex of the parabolic height function, which gives the time at which the peak height occurs, then calculating the height at that time.

Step-by-step explanation:

To find out when splashdown occurs for the rocket, we need to determine when the height function H(t) = -4.9t^2 + 268t + 325 equals zero. This is because splashdown will occur when the height above sea-level reaches zero again after launch. To solve for t, we can use the quadratic formula, which states that for an equation ax^2 + bx + c = 0, the solutions for x are:

x = (-b ± √(b^2 - 4ac)) / (2a).

In this case, a = -4.9, b = 268, and c = 325. To avoid physical impossibilities, we will disregard the negative value for t as it would imply a time before the launch. Using this formula will give us the time when the rocket splashdown occurs.

To find the peak height, we need to determine the vertex of the parabola described by the height function. The time at which the peak occurs is given by t = -b/(2a), and we can then substitute this value back into the height function to find the maximum height above sea-level.