H(t)= -16t^2 + 87t. find the average rate of change of the height of the ball from 2 to 4.5 seconds

  -17 ft/s

Step-by-step explanation:

The average rate of change is the difference in height on the interval, divided by the difference in time.

  m = (h(4.5) -h(2))/(4.5 -2) . . . . . . . . definition of rate of change on [2, 4.5]

  h(4.5) = (-16·4.5 +87)(4.5) = 67.5

  h(2) = (-16·2 +87)(2) = 110

Then ...

  m = (67.5 -110)/(2.5) = -17

The average rate of change on the interval from 2 to 4.5 seconds is -17 ft/s.

Check

The slope of the curve is given by its derivative:

  h'(t) = -32t +87

The midpoint of the interval is (2+4.5)/2 = 3.25.

The slope at the midpoint is ...

  h'(3.25) = -32(3.25) +87 = -17 . . . . average rate of change on [2, 4.5]

For a quadratic function. the average rate of change on an interval is the slope of the function at the midpoint of the interval. This calculation confirms that.

there is no

i do not know this sorry : (