A 15-g sample of radioactive iodine decays in such a way that the mass remaining after t days is given by
m(t) = 15e−0.051t,
where
m(t)
is measured in grams. After how many days is there only 1 g remaining?
just find t where
15 e^(-0.051t) = 1
To find the number of days when there is only 1 gram remaining, we need to solve the equation:
m(t) = 1
Substituting the given exponential decay function, we have:
15e^(-0.051t) = 1
Now, let's solve this equation step by step to find the value of t:
Divide both sides by 15:
e^(-0.051t) = 1/15
Take the natural logarithm on both sides:
ln(e^(-0.051t)) = ln(1/15)
Using the logarithmic property, we can bring down the exponent:
-0.051t ln(e) = ln(1/15)
Since ln(e) is equals to 1:
-0.051t = ln(1/15)
Now, divide both sides by -0.051:
t = ln(1/15) / -0.051
Using a calculator, we can evaluate the right side of the equation:
t ≈ 27.041
Therefore, after approximately 27.041 days, there will be only 1 gram remaining.