hitting a golf ball

Question

The height of a golf ball is given by h(t) = -16t² +48t, where t is the time in seconds and h(t) is the height in feet. 

 

A) Why does it make sense that there is no "c" value for this equation in the context of the problem

B) At what time is the golf ball on the ground?

C) At what time is the gold ball at its highest point?

D) How high does the golf ball go? At what time is it?

E) What domain and range values make sense in the context of this situation?

Answer 1

A. c represents the height of the golf ball at the initial time, t=0s.  Since the ball initially starts on the ground, corresponding to height = 0feet, c = 0

 

B. Since h(t) = height above ground, ball on ground means that h(t) = 0. 

   -16t² + 48t = 0

    48t = 16t²

    3t = t²

        The solution to this is t=0s (initial time), and t=3s

 

C. Since the height of the ball is a parabola, the time it reaches its maximum will be halfway between the two times that the ball is on the ground.  These two times were calculated in part B: 0s and 3s.  

So the time when the ball is at its highest point is (0s+3s)/2 = 1.5s 

 

D. The time that the golf ball is at its highest point was calculated in part C to be 1.5s.   The actual height it reaches can be found by putting in t = 1.5s in the equation for h(t)

highest point = h(1.5) = -16 * 1.5² + 48*1.5 = 36 feet

 

E. The domain that makes sense is the values of t that give an answer that is physically real.  At time t=0s the ball begins to rise, with a height given by h(t).  This function is only valid for times greater or equal to t=0s.  Since the ball cannot go underground, once the ball lands on the ground again, which happens at t=3s, it remains on the ground and its height is no longer represented by h(t).  

So, the domain that makes sense is t = 0s to t=3s.  

 

  The range that makes sense is the minimum and maximum height the ball gets to during this time period.  Obviously, the minimum is h(t) = 0ft, since it starts at ground level.  The highest point was calculated in part D to be 36 feet.

So the range that makes sense is h(t) = 0ft to 36ft