For the radioactive decay model Q(t)=200(.86^t), divide the range of t values over which the quantity decrease to approximately 1/4 of its orginial amount into three equal segments. For each segment show that the ratio of the average rate of change of Q to the average value of Q over th interval is in fact bearly constant.......I am really lost when it comes to this problem....I am really not sure where to start....should I pluge 1/4 in for t?....and if so where do I go from there? Thanks.
Sketch a graph of Q vs. t. The basic points are (0, 200) and (1, 172); be sure you understand why. And the graph approaches the positive x axis as an asymptote.
The initial value is 200.
According to your graph how long does it take to fall to about 1/4 of that value?
If for example you found that it took from t = 0 to t = 36 to fall to about 1/4 of its original value (it doesn't take that long, but suppose it did), you would divide that interval into three equal segments, one from t = 0 to t = 12, one from t = 12 to t = 24 and one from t = 24 to t = 36.
For each interval you would find the rate of change of Q with respect to t: change in Q / change in t.
For each interval you would also find the approximate average value of Q. From finding the rates of change you will know the value of Q at the beginning and end of each interval. So you can reasonably estimate the average value of Q for each interval--the way the graph is shaped, the average value will be a little closer to the minimum value than the maximum value on each interval.
Once you have determined the average rate of change and the average value for each of the three intervals, you can find the ratio of the average rate of change to the average value for each of the three intervals. The ratio of A to B is A / B.