Question

A rocket is launched at t=0 seconds. Its height, in meters above sea-level, as a function of time is given by h(t)=−4.9t^2+220t+311. At what time does the rocket reach its maximum height? (Round your answer to 2 decimal places.) The rocket reaches its maximum height after ____ seconds. At what height does the rocket reach its maximum height above the water? (Round your answer to 2 decimal places.) The rocket reaches its maximum height at ______ meters above sea-level.

a. 22.45 seconds, 1024.76 meters
b. 18.67 seconds, 511.32 meters
c. 20.93 seconds, 731.24 meters
d. 24.11 seconds, 892.15 meters

Answer 1

The rocket reaches its maximum height at 22.45 seconds and the maximum height is 1024.76 meters above sea-level, which corresponds to option (a).

Step-by-step explanation:

The rocket launched and described by the function h(t) = -4.9t2 + 220t + 311 will reach its maximum height when its first derivative, which represents the velocity, equals zero. The first derivative of the height function with respect to time t is h'(t) = -9.8t + 220. Setting this equal to zero to find the time when the rocket reaches its maximum height gives us:

0 = -9.8t + 220
t = 220 / 9.8
t = 22.45 seconds

To find the maximum height, we substitute this time back into the height function:

h(22.45) = -4.9(22.45)2 + 220(22.45) + 311
The rocket reaches its maximum height at 1024.76 meters above sea-level.

Therefore, the correct answer is option (a): The rocket reaches its maximum height after 22.45 seconds at a height of 1024.76 meters above sea-level.