Answer to Question #105322 in Calculus for Kaitlyn Pankoe

Question

A certain radioactive isotope has a half-life of 15 days. If one is to make a table showing the half-life decay of a sample of this isotope from 32 grams to 1 gram; list the time (in days, starting with t=0) in the first column and the mass remaining (in grams) in the second column, which type of sequence is used in the first column and which type of sequence is used in the second column?

Answer 1

What does half life means?

Half life of a substance means after each certain time interval, half quantity of substance decay and half remains. For example if a substance have half life of t days, after each t days, half of the substance will decay and only half quantity of substance will remain.

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If the sample has initial quantity = 32 grams and half life is 15 days

then after each 15 days , half of its quantity remains

i.e. after first 15 days = 32/2 = 16 grams will remain

after another 15 days (i.e. after 2 half life = total 15+15 =30 days from start) = 16/2 = 8 grams will remain

after another 15 days (i.e. after 3 half life = total 15*3 =45 days from start) = 8/2 grams will remain

Similarly it can be summarize as follow until 1 gram of sample remain.

Here time (t) = 0 represents the initial quantity.

Now looking on first column of table, we can observe that the time is increasing by 15 days as we go from top to bottom.

i.e. difference between two rows is 15 days

(15-0) = (30-15)=(45-30)=(60-45)=(75-60) = 15

i.e. common differences are equal

This shows the Arithmetic sequence has been used in first column

Similarly in second row, we can see that the quantity is getting half each time as we go from top to bottom

i.e. ratio of next row to first row is = 1/2

(16/32)=(8/16)=(4/8)=(2/4)=(1/2) = 1/2

i.e. common ratios are constant

This shows Geometric sequence has been used in second column.