The half-life of a certain radioactive sub- stance is 8 hrs. There are 5 grams present initially. Which of the following gives the best approximation when there will be 1 gram remaining?
Would you use the population growth equation?
you would need the decay equation
I would use
amount = 5(1/2)^(t/8), where t is in hours
1 = 5(.5)^(t/8)
.2 = .5^(t/8)
take log of both sides and use log rules
log .2 = (t/8)log .5
t/8 = log .2/log .5 = 2.3219....t = 18.57 hours or appr 3 hours
check:
after 8 hours you would 2.5 g left
after 16 hours you would have 1.25 g left
3 hours to go to get to 1 g ?? , seems reasonable
To find the time at which there will be 1 gram remaining, we can use the equation for exponential decay:
N(t) = N₀ * (1/2)^(t/h)
Where:
N(t) is the amount of the substance at time t
N₀ is the initial amount of the substance
t is the time that has passed
h is the half-life of the substance
In this case, N₀ = 5 grams and we want to find the time t when N(t) = 1 gram:
1 = 5 * (1/2)^(t/8)
We can solve for t by taking the logarithm of both sides:
log₁/₂(1) = log₁/₂(5) + (t/8) * log₁/₂(1/2)
Simplifying, we have:
0 = log₁/₂(5) + (t/8) * (-1)
By rearranging this equation, we can solve for t:
(t/8) = -log₁/₂(5)
t = 8 * (-log₁/₂(5))
Using a calculator, we find:
t ≈ 8 * (-2.3219)
t ≈ -18.5753
Since time cannot be negative, we discard the negative result and conclude that there will be approximately 1 gram remaining after 18.5753 hours.
Therefore, the best approximation when there will be 1 gram remaining is approximately 18.6 hours.
To determine when there will be 1 gram remaining of the radioactive substance, we can use the half-life concept. The half-life of a substance is the time it takes for half of the initial amount to decay.
In this case, the half-life is 8 hours. This means that after 8 hours, half of the substance will have decayed, leaving us with 2.5 grams.
To find out when there will be 1 gram remaining, we need to continue applying the concept of half-life.
After the first half-life (8 hours), we have 2.5 grams remaining. After the second half-life (another 8 hours), we'll have half of 2.5 grams, or 1.25 grams.
So, we can approximate that after approximately 16 hours, there will be 1.25 grams remaining. Since we want to find when there will be 1 gram remaining, we know that we need to wait a bit longer than 16 hours.
Based on the given options, the best approximation would be either 17 or 18 hours. However, since we know it will be a bit longer than 16 hours, the best choice would be 18 hours.