Question

For the following exercises, use this scenario: A tumor is injected with 0.5 grams of Iodine-125, which has a decay rate of 1.15% per day.

A scientist begins with 250 grams of a radioactive substance. After 250 minutes, the sample has decayed to 32 grams. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest minute, what is the half-life of this substance?

Answer 1

Consider that a scientist begins with 250 grams of a radioactive substance and the sample decays to 32 grams after 250 minutes.

 

The objective is to find the nearest minute.

The continuous growth formula is,

A(t) = A0ert

 

Suppose that amount of the substance remaining afterminutes is given by,

A(t) = 250ert ….. (1)

 

Substitute the values of A = 32 and t = 250 in equation (1).

        32 = 250er(250)

er(250) = 32/250

er(250) = 0.128

 

Apply natural logarithm on both sides.

ln{er(250)} = ln(0.128)

   250rln(e) = ln(0.128)

                 r = ln(0.128)/250      {ln(e) = 1}

                   ≈ -0.008

 

Plug r ≈ -0.008 in equation A(t) = 250ert.

    A(t) = 250e-0.008t

 

Thus, the amount of the substance remaining after t minutes is,

A(t) = 250e-0.008t….. (2)

 

Find the nearest minute when the half -life of the substance.

Substitute A(t) = 250/2 in equation (2).

        250/2 = 250e-0.008t

            1/2 = e-0.008t

    e-0.008t = 0.5

ln(e-0.08t) = ln(0.5)

-0.008tln(e) = ln(0.5)

   -0.008t(1) = ln(0.5)   {ln(e) = 1}

                  t = ln(0.5)/-0.008

                    ≈ 87

 

Thus, the half life of the substance is 87 minutes.

Thus, the half life of the substance is 87 minutes.