Tag Archives: exponential decay

Ordinary Differential Equations by Tenenbaum & Pollard

ODE by Tenenbaum & Pollard, Exercise 1, Problem 3

August 2, 2014 etoix Leave a comment

Problem

The following item appeared in a newspaper. “The expedition used the carbon-14 test to measure the amount of radioactivity still present in the organic material found in the ruins, thereby determining that a town existed there as long ago as 7000 B.C.” Using the half-life figure of $C^{14}$ as given in the text, determine the approximate percentage of $C^{14}$ still present in the organic material at the time of the discovery.

Solution

Exponential decay is defined by $y = Ce^{kt}$ where C is what we start with, k is the rate of growth or decay, t is time, and y is what we got after time t. The text derived -0.000124 for the half-life rate.

$k = -0.000124$

$C= 100$

$t = 9014$

Solve for y.

$y = 100 e^{-0.000124 9014}$

By calculator $y = 32.7%$ .

If the time of discovery were 2014, the percentage remaining would be 32.7%.

Ordinary Differential Equations by Tenenbaum & Pollard

ODE by Tenenbaum & Pollard, Exercise 1, Problem 2

August 1, 2014 etoix 1 Comment

Problem

Assume that the half life of the radium in a piece of lead is 1600 years. How much radium will be lost in 100 years?

Solution

Exponential decay is defined by $y = Ce^{kt}$ where C is what we start with, k is the rate of growth or decay, t is time, and y is what we got after time t.

$y = 50$

$C = 100$

$t = 1600$

Solve for k.

$50 = 100 e^{1600 k}$

$\frac{1}{2} = e^{1600 k}$

$1600 k = \ln{\frac{1}{2}}$

$k = \frac{ \ln{\frac{1}{2}} }{ 1600 }$

Now that we know k, solve for $t = 100$

$y = 100 e^{ \frac{ \ln{\frac{1}{2}} }{ 16 } }$

By calculator $y = 95.76$ .

y is what we have left but the question asks for how much we lost.

$100 - 95.76 = 4.24$

After 100 years, we will have lost 4.2% of the radium.

Ordinary Differential Equations by Tenenbaum & Pollard

ODE by Tenenbaum & Pollard, Exercise 1, Problem 1

July 31, 2014 etoix Leave a comment

Problem

The radium in a piece of lead decomposes at a rate which is proportional to the amount present. If 10 percent of the radium decomposes in 200 years, what percent of the original amount of radium will be present in a piece of lead after 1000 years?

Solution

Exponential decay is defined by $y = Ce^{kt}$ where C is what we start with, k is the rate of growth or decay, t is time, and y is what we got after time t.

$y = 90$