Question

A scientist begins with 100 milligrams of a radioactive substance that decays exponentially. After 35 hours, 50 mg of the substance remains. How many milligrams will remain after 54 hours?

Answer 1

The objective is to find the amount of radioactive substance that remains after 54 hours.

The formula for the continuous growth or decay is,

A(t) = aert

 

The initial amount of radioactive substance is a = 100 and A(t) = 50. The elapsed time is t = 35 hours. First, find the value of r as follows:

Substitute the known values in A(t) = aert

        50 = 100(e35r)

50/100 = e35r

       0.5 = e35r

 

Take natural log on both sides as follows:

ln(0.5) = ln(e35r)

         r = ln(0.5)/35

           = -0.02

Hence,

        r = -0.02

 

Therefore, the exponential function decays by 2%.

 

The amount of milligram remains after 54 hours is determined as follows:

A(t) = 100{e54(-0.02)}

       = 100e-1.08

       = 33.96

 

Therefore, the amount of milligram remains after 54 hours is 33.96 milligrams.

Therefore, the amount of milligram remains after 54 hours is 33.96 milligrams.