Element X is a radioactive isotope such that every 86 years, its mass decreases by half. Given that the initial mass of a sample of Element X is 730 grams, how much of the element would remain after 9 years, to the nearest whole number?
Answer is 679
730 * (1/2)^(9/86)
=679
Well, radioactive decay is quite a weighty matter, isn't it? Let's see if we can untangle this atomic conundrum.
If Element X loses half its mass every 86 years, after 9 years, we can calculate how many half-lives occur. Divide 9 by 86 and round down because this is a decay party – we're losing mass, not gaining it. So, we have about 0.1047 half-lives.
Now, for each half-life, the mass decreases by half. But since we have a fraction of a half-life, things get a bit tricky. If we consider this as a continuous decay model, we can multiply the initial mass by 0.5 raised to the power of the number of half-lives (approximately 0.1047).
Applying this formula, after 9 years, we can estimate that around 711 grams of Element X would remain. Oh, don't worry, it's all part of the radioactive magic trick! Poof, and half the mass is gone!
To solve this problem, we can use the formula for exponential decay:
Final mass = Initial mass × (1/2)^(time / half-life)
Given that the half-life of Element X is 86 years, and the initial mass is 730 grams, we can substitute these values into the formula:
Final mass = 730 × (1/2)^(9 / 86)
Calculating this expression gives us:
Final mass = 730 × (1/2)^(0.1047)
Using a calculator, we find:
Final mass ≈ 730 × 0.9050
Final mass ≈ 661.7 grams
Rounding this to the nearest whole number, the amount of Element X remaining after 9 years is approximately 662 grams.
To solve this problem, we need to calculate how many times the mass of Element X decreases by half over a period of 9 years.
We know that the half-life of Element X is 86 years, which means that after every 86 years, its mass decreases by half.
First, we need to find out how many half-life periods are there in 9 years. We can do this by dividing 9 by 86:
9 years / 86 years = 0.1047
This tells us that the sample of Element X has undergone approximately 0.1047 half-life periods in 9 years.
Next, we need to calculate the remaining mass of Element X after a fraction of a half-life period. To do this, we raise 0.5 to the power of the fraction of a half-life period:
Remaining mass = Initial mass * (0.5)^(fraction of a half-life period)
In this case, the initial mass is 730 grams and the fraction of a half-life period is 0.1047. Plugging these values into the formula, we get:
Remaining mass = 730 grams * (0.5)^(0.1047)
Calculating this expression gives us:
Remaining mass ≈ 687 grams
Therefore, after 9 years, approximately 687 grams of Element X would remain, when rounded to the nearest whole number.
The half-life is 86 years, so the decay function is
730 * (1/2)^(t/86)
So plug in t=9