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.