a need help for this question

Question

A baseball is hit so that its height in feet, t seconds after it is hit, can be modeled by the function h(t) = –8t^2 + 37t – 3. A hawk takes off from its nest 28 feet above the ground at the same time that the player hits the baseball. The hawk’s flight can be modeled by the function h(t) = 28 − t. After how many seconds do the hawk and baseball first achieve the same height? What is that height?

 

a. 1.05 seconds and 26.95 feet off the ground
b. 2.31 seconds and 39.78 feet off the ground
c. The bird and the ball will not reach the same height above the ground.
d. 3.7 seconds and 24.3 feet off the ground

Answer 1

Set the two height functions equal to each other, solve for time t, and substitute the result for t into either height function to obtain the height:

 

-8t² + 37t - 3 = 28 - t ⇒

 

-8t² + 38t - 31 = 0

 

Solve this quadratic equation for t.  You will get two solutions. One will be negative, which you can discard.  The other will be +3.7 seconds.  That is the time they both reach the same height.  Substitute this value of t into either height equation to get h = 24.3 feet.