Q:

If a stone is dropped from a height of 256 ​ft, then its height​ (in feet) above the ground is given by the function ​h(t)equalsnegative 16 t squared plus 256 where t is time​ (in seconds). To get an idea of how fast the stone is traveling when it hits the​ ground, find the average rate of change of the height on each of the time intervals ​[0,4​], ​[1,4​], ​[3.9​,4​], ​[3.99​,4​], and ​[3.999​,4​].

Accepted Solution

A:
Answer:For ​[0,4​] : -64For ​[1,4​] : -80For ​[3.9,4​] : -126.4For ​[3.99,4​] : -127.84For ​[3.999,4​] :  -127.984Step-by-step explanation:First we establish that we have the next function:                                         h(t) = -16t² + 256Now, to find the average rate of change we have to define the change in the output of the function and the change in the input of the function:   Average rate of change = Change in the output / Change in the inputWhich in this case will be represented by:                         Average rate of change = h(t2) - h(t1) / t2 - t1So now we resolve the function for each time interval and calculare the average rate of change:For ​[0,4​] : h(0) = -16(0)² + 256 = 256 h(4) = -16(4)² + 256 = 0Average rate of change = [0 - 256] / [4 - 0] = -64For ​[1,4​] : h(1) = -16(1)² + 256 = 240 h(4) = -16(4)² + 256 = 0Average rate of change = [0 - 240] / [4 - 1] = -80For ​[3.9,4​] : h(3.9) = -16(3.9)² + 256 = 12.64 h(4) = -16(4)² + 256 = 0Average rate of change = [0 - 12.64] / [4 - 3.9] = -126.4For ​[3.99,4​] : h(3.99) = -16(3.99)² + 256 = 1.2784 h(4) = -16(4)² + 256 = 0Average rate of change = [0 - 1.2784] / [4 - 3.99] = -127.84For ​[3.999,4​] : h(3.999) = -16(3.999)² + 256 = 0.127984 h(4) = -16(4)² + 256 = 0Average rate of change = [0 - 0.127984] / [4 - 3.999] = -127.984