How do I set-up/solve this problem? Please help me.
The half-life of the radioactive isotope gallium-70 is 20 minutes. If the initial amount of gallium-70 is 25 milligrams, when will 10 milligrams be left?
can you solve
10 = 25(.5)^(t/20) where t will be in minutes ?
This is what I got..
25(1/2)^(t/20)=10
(1/2)^(t/20)=10/25=2/5
(1/2)^(t/20)=2/5
t/20=log(2/5)/log(1/2)
t=20(log(2/5)/log(1/2))
t=26.44
Is this correct?
correct, good job.
To set up and solve this problem, you need to use the formula for exponential decay:
N(t) = N₀ * (1/2)^(t / t₁/₂)
Where:
N(t) is the amount of the isotope at time t
N₀ is the initial amount of the isotope
t₁/₂ is the half-life of the isotope
t is the time elapsed
In this problem, you are given:
N₀ = 25 milligrams (initial amount)
t₁/₂ = 20 minutes (half-life)
We need to find the time t when there will be 10 milligrams left.
Now, let's substitute the given values into the equation and solve for t:
10 = 25 * (1/2)^(t / 20)
To isolate the variable, we can divide both sides of the equation by 25:
(10/25) = (1/2)^(t / 20)
Simplifying:
0.4 = (1/2)^(t / 20)
To remove the exponent, we can take the logarithm (base 2) of both sides:
log₂(0.4) = log₂((1/2)^(t / 20))
Using the property of logarithms, we can bring down the exponent:
log₂(0.4) = (t / 20) * log₂(1/2)
Now, solve for t by multiplying both sides of the equation by 20 and dividing by log₂(1/2):
t = 20 * (log₂(0.4) / log₂(1/2))
Now you can use a calculator to evaluate the right side of the equation. The answer will give you the time t in minutes when there will be 10 milligrams of gallium-70 left.