Question 104·Medium·Nonlinear Functions
Carbon-14 has a half-life of about 5,730 years. If a freshly created sample contains 80 grams of Carbon-14, after approximately how many years will only 10 grams remain?
For half-life questions, avoid complicated formulas at first and think in terms of repeated halving. Start from the initial amount and keep halving it once per half-life until you reach (or pass) the target amount, counting how many halvings you used. Then multiply the number of half-lives by the given half-life duration to get the total time. When answer choices are multiples of the half-life (as on the SAT), you can quickly test each by tracking how many halvings they represent.
Hints
Think in steps of one half-life
Ask yourself: after 5,730 years (one half-life), what fraction of the original 80 grams will remain, and how many grams is that?
Continue halving until you reach 10 grams
After the first halving, halve the new amount again after another 5,730 years, and keep going until you reach 10 grams. Count how many halvings (half-lives) that takes.
Turn half-lives into years
Once you know how many half-lives are needed, multiply that number by 5,730 to find the total time in years.
Desmos Guide
Graph the decay function and the target amount
In Desmos, enter the two expressions on separate lines:
y = 80*(1/2)^(x/5730)(this models the amount of Carbon-14 after x years)y = 10(this is the target amount) Adjust the viewing window so you can see where these two graphs intersect.
Find the time when the amount is 10 grams
Click on the intersection point of the two graphs. The x-coordinate of this point is the number of years it takes for the sample to decay from 80 grams to 10 grams. Compare that x-value with the answer choices and select the closest match.
Step-by-step Explanation
Understand what half-life means
A half-life is the time it takes for a substance to decrease to half of its current amount.
So every 5,730 years, the amount of Carbon-14 is multiplied by (cut in half). We start with 80 grams and apply this halving repeatedly.
Track the mass through each half-life
Start from 80 grams and halve it once every 5,730 years:
- After 1 half-life (5,730 years): grams.
- After 2 half-lives (11,460 years): grams.
- After 3 half-lives: grams.
So it takes 3 half-lives for the sample to go from 80 grams down to 10 grams.
Convert number of half-lives to years and choose the answer
Each half-life is 5,730 years, and we need 3 half-lives:
So it will take about 17,190 years for the sample to decrease from 80 grams to 10 grams. The correct choice is C) 17,190.