assume the substance has a half-life of 12 years and the initial
amount is 133 grams. How long will it be until only 40 % remains?
1(1/2)^(t/12) = .4
2^(-t/12) = .4
take log of both sides
(-t/12) log 2= log .4
t = -12 log .4/log2 = appr. 15.9 years.
To determine how long it will take for only 40% of the substance to remain, we need to use the concept of half-life and exponential decay.
The half-life is the amount of time it takes for half of the substance to decay. In this case, the half-life is 12 years, meaning that after 12 years, half of the substance will decay, and only 50% will remain.
Now, we need to find out how many half-lives it will take for only 40% to remain. Since each half-life reduces the amount by half, we can calculate the number of half-lives using the following formula:
Number of Half-lives = (log(remaining amount) - log(final amount)) / log(0.5)
Let's calculate it step by step:
1. Convert the given percentage (40%) to a decimal: 40% = 0.40
2. Calculate the final amount using the initial amount (133 grams) and the remaining percentage:
Final amount = 133 grams * 0.40 = 53.2 grams
Now, using the formula mentioned above:
Number of Half-lives = (log(53.2) - log(133)) / log(0.5)
To get the logarithm of a number, we use the base 10 logarithm (log), which is usually available on calculators or math software.
Number of Half-lives = (log(53.2) - log(133)) / log(0.5) ≈ 0.217
Since it's not possible to have a fraction of a half-life, we round up to the next whole number.
Number of Half-lives = 1
Therefore, it will take approximately one half-life, which is equal to 12 years, for only 40% of the substance to remain.