X S2.0.2 A rocket is fired upward with an initial velocity v of 80 meters per second. The quadratic function S(t) = -52 + 80t can be used to find
the heights of the rocket, in meters, at any time t in seconds. Find the height of the rocket 8 seconds after it takes off. During the
course of its flight, after how many seconds will the rocket be at a height of 290 meters?

9514 1404 393

Step-by-step explanation:

To find the height at 8 seconds, evaluate the formula for t=8.

  S(t) = -5t^2 +80t

  S(8) = -5(8^2) +80(8) = -320 +640 = 320

The height of the rocket is 320 meters 8 seconds after takeoff.

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To find the time to 290 meters height, solve ...

  S(t) = 290

  290 = -5t^2 +80t

  -58 = t^2 -16t . . . . . . . divide by -5

  6 = t^2 -16t +64 . . . . . complete the square by adding 64

  ±√6 = t -8 . . . . . . . . . take the square root

  t = 8 ±√6 ≈ {5.551, 10.449}

The rocket is at 290 meters height after 5.6 seconds and again after 10.4 seconds.