Element X decays radioactively with a half life of 5 minutes. If there are 540 grams of Element X, how long, to the nearest tenth of a minute, would it take the element to decay to 252 grams?
540(1/2)^(x/5) = 252
now just solve for x
Well, Element X sure knows how to shred weight! Let's do some calculations, shall we? If the half-life of Element X is 5 minutes, it means that every 5 minutes the amount of Element X is cut in half.
So, to go from 540 grams to 252 grams, we need to determine how many half-lives it takes.
First, subtract 252 from 540, which gives us 288 grams. Now, divide 288 by half of 540 to find out how many half-lives it takes. We get 288/(540/2) = 1.0666667 half-lives.
Now, to find the time it takes for 1.0666667 half-lives, we simply multiply by the half-life of 5 minutes.
1.0666667 * 5 = 5.3333335 minutes.
To the nearest tenth of a minute, it would take approximately 5.3 minutes for Element X to decay from 540 grams to 252 grams. Don't blink, though, you might miss it!
To determine the time it would take for Element X to decay from 540 grams to 252 grams, we need to find the number of half-lives that occur.
1. Start by finding the difference in the initial and final masses of Element X: 540 g - 252 g = 288 g.
2. Next, calculate the number of half-lives by dividing the difference in mass by the initial mass of one half-life: 288 g / 540 g = 0.5333.
3. Since each half-life is 5 minutes, multiply the number of half-lives by 5 to find the total time in minutes: 0.5333 x 5 = 2.667 minutes.
Therefore, it would take approximately 2.7 minutes (to the nearest tenth of a minute) for Element X to decay to 252 grams.
To solve this problem, we need to use the equation for radioactive decay:
N = N₀ * (1/2)^(t/h)
Where:
N = final amount of the element
N₀ = initial amount of the element
t = time elapsed
h = half-life of the element
In this case, we know the initial amount (N₀ = 540 grams), and we want to find the time elapsed (t) when the final amount (N) is 252 grams. The half-life (h) is given as 5 minutes.
Let's plug in the values into the equation and solve for t:
252 = 540 * (1/2)^(t/5)
To isolate the exponent, we can take the logarithm (base 1/2) of both sides:
log₁/₂(252/540) = t/5
To find the value of the logarithm, we can use the logarithmic identity:
logₐ(b) = log(c) / log(a)
Rearranging the equation using this identity, we have:
t/5 = log(252/540) / log(1/2)
Now, let's calculate the logarithms:
t/5 = log(252/540) / log(1/2)
t/5 = -0.531478917 / -0.693147181
Dividing on both sides and multiplying by 5, we get:
t ≈ 5 * (-0.531478917 / -0.693147181)
t ≈ 3.846313981 minutes
Therefore, it would take approximately 3.8 minutes (to the nearest tenth of a minute) for the element to decay to 252 grams.