If you have 10,000 grams of a substance that decays with a half-life of 14 days, then how much will you have after 70 days?
Show your work and round your answer to the nearest whole number.
After 1 half-life (14 days), the amount of substance remaining is half of the original amount:
10,000 grams / 2 = 5,000 grams
After 2 half-lives (28 days), the amount remaining is half of the amount remaining after 1 half-life:
5,000 grams / 2 = 2,500 grams
After 3 half-lives (42 days), the amount remaining is half of the amount remaining after 2 half-lives:
2,500 grams / 2 = 1,250 grams
After 4 half-lives (56 days), the amount remaining is half of the amount remaining after 3 half-lives:
1,250 grams / 2 = 625 grams
After 5 half-lives (70 days), the amount remaining is half of the amount remaining after 4 half-lives:
625 grams / 2 = 312.5 grams
Rounding to the nearest whole number, you would have approximately 313 grams remaining after 70 days.
or, done in a more direct way:
amount = 10,000(1/2)^(70/14)
= 10000(1/2)^5
= 10000(1/32) = appr 313
Yes, that is another way to calculate it directly using the formula A = A0(1/2)^(t/T), where A0 is the initial amount, t is the time elapsed, and T is the half-life. This formula gives the same result as the step-by-step approach.
To determine the amount of substance remaining after a given time period, we can use the formula:
N = N₀ * (1/2)^(t/t₁/₂)
Where:
N = final amount of substance
N₀ = initial amount of substance
t = time elapsed
t₁/₂ = half-life of the substance
In this case, N₀ is 10,000 grams, t is 70 days, and t₁/₂ is 14 days. Let's substitute these values into the formula and calculate:
N = 10,000 * (1/2)^(70/14)
To simplify, we can divide the exponent:
N = 10,000 * (1/2)^5
Now, let's calculate the value inside the parentheses:
(1/2)^5 = (1/32)
Substituting this back into the formula:
N = 10,000 * (1/32)
Now, we can calculate the final amount:
N = 312.5
Rounding this to the nearest whole number, we get:
N ≈ 313 grams
Therefore, after 70 days, you will have approximately 313 grams of the substance remaining.