The height of a ball dropped from a tall building is modeled by the equation d(t) = 16t2 where d equals the distance traveled at time t seconds and t equals the time in seconds. What does the average rate of change of d(t) from t = 2 to t = 5 represent?

To find the average rate of change of d(t) from t = 2 to t = 5, we need to find the change in distance (d) divided by the change in time (t).

The distance at time t = 2 seconds is given by d(2) = 16(2)^2 = 16(4) = 64.
The distance at time t = 5 seconds is given by d(5) = 16(5)^2 = 16(25) = 400.

Therefore, the change in distance is d(5) - d(2) = 400 - 64 = 336.

The change in time is t(5) - t(2) = 5 - 2 = 3.

Now, we can calculate the average rate of change:

Average rate of change = Change in distance / Change in time
= 336 / 3
= 112.

Therefore, the average rate of change of d(t) from t = 2 to t = 5 is 112.

To find the average rate of change of a function, we need to calculate the change in the function's values divided by the change in the independent variable. In this case, the function is d(t) = 16t^2, and we want to find the average rate of change from t = 2 to t = 5.

Step 1: Calculate the value of d(t) at t = 2 and t = 5.
- Substitute t = 2 into the equation: d(2) = 16(2)^2 = 16(4) = 64.
- Substitute t = 5 into the equation: d(5) = 16(5)^2 = 16(25) = 400.

Step 2: Calculate the change in d(t) and the change in t.
- Change in d(t) = d(5) - d(2) = 400 - 64 = 336.
- Change in t = t2 - t1 = 5 - 2 = 3.

Step 3: Calculate the average rate of change.
- Average rate of change = (change in d(t))/(change in t) = 336/3 = 112.

Therefore, the average rate of change of d(t) from t = 2 to t = 5 is 112, which represents the average change in distance traveled per second during that time interval.

It represents the average speed during that interval or in other words the distance traveled during that interval/ 3 seconds

d(5) = 16 * 25 = 400
d(2) = 16*4 = 64
distance during those three seconds = 400-64 = 336 feet
336 ft/3 s = 112 ft/s = average speed for that interval

By the way they mean distance fallen = 16 t^2
the height h = building height - 16 t^2

hgy