Applications of Exponential Decay

Carbon-14 dating

The half-life of Carbon 14 ($^14\C$) is 5730 years. This means that every 5730 years half of the remaining $^14\C$ in an organism disintegrates. For example, if there are 12 grams of $^14\C$ to start with, then after 5730 years there will be 6 grams remaining, and after the second 5730 years there will be half of the 6 grams, or 3 grams, remaining. Once something dies no more $^14\C$ is absorbed, so we can tell how long it has been dead by comparing how much $^14\C$ is currently in it to how much $^14\C$ was in it when it died.

Let $C$ be the amount of $^14\C$ left in an organism after $t$ years, if it had 1 gram of $^14\C$ when it died. We want to find an equation relating $C$ and $t$.

As time passes, will there be more or less $^14\C$ remaining in the organism?
As $t$ increases, will $C$ increase or decrease?
After 5730 years, there will be 0.5 grams of $^14\C$ left. How much $^14\C$ will be left after 5730 more years (11460 total)? grams
How much $^14\C$ will be left in 17190 years (or $3 ⋅ 5730$)? grams

Every 5730 years, the amount of $^14\C$ is multiplied by $$1/2$$. This means that, every $t$ years, the amount of $^14\C$ is multiplied by $$(1/2)^{t∕5730}$$.

Notice that your last two answers have been plotted on the grid to the left. Click to see the graph of $$C = (1/2)^{t∕5730}$$. Does that graph pass through the plotted points?
Click and drag on the graph to move the vertical bar so that $t$ has the value 10000. What value does $C$ have when $t$ is 10000? Give your answer to four decimal places.

We actually want to use this equation to find out how long ago something died (like a fossil), so let’s look at it in a slightly different way. Suppose that you know that the $^14\C$ in a fossil is 0.1629 grams, and the amount that was in the organism when it died was 1.0 gram. Drag the vertical bar on the grid until $C$ has the value 0.1629. You see that $t = 15000$. This means that the organism died 15000 years ago.

Use this method (dragging the vertical bar to the given $C$ value) to find how long ago that organism died if there’s now 0.2643 grams of $^14\C$. years
How long ago did it die if the current $^14\C$ amount is 0.003008 grams? years

Cooling and the equation $y = m ⋅ a^{-t} + k$

A room with several large windows is normally kept at a temperature of 20 degrees Celsius (68 degrees Fahrenheit). The heat in the room is turned entirely off while the temperature outside is 5 degrees Celsius (41 degrees Fahrenheit). It’s then found that the temperature $y_1$ in the room $t$ hours later, in degrees Celsius, obeys the equation $y_1 = 15 ⋅ 2.33^{-t} + 5$. This equation has been graphed on the grid to the left.

When $t=0$ (that is, at the moment the heat is turned off), what is the temperature in the room in degrees Celsius?

What is the temperature in the room in degrees Celsius 3.5 hours after the heat is turned off, to three decimal places? (Hint: Click or tap inside the grid to find the answer.)
Approximately how long does it take for the temperature in the room to drop below 15 degrees Celsius? hours
Approximately how long does it take for the temperature to drop below 10 degrees Celsius? hours

In the room from tempQn1, the glass in the windows is replaced by double-glazed glass, which lets less heat through. It’s then found that, if the heat is turned off while the temperature outside is 5 degrees Celsius, the temperature $y_2$ in the room $t$ hours later obeys the equation $y_2 = 15 ⋅ 1.39^{-t} + 5$.

What is the temperature in this room 3.5 hours after the heat is turned off, to three decimal places? degrees Celsius
Approximately how long does it take for the temperature in the room to drop below 15 degrees Celsius? hours
Approximately how long does it take for the temperature to drop below 10 degrees Celsius? hours

The two equations for temperature are both graphed on the grid to the left. The equation for the temperature with ordinary windows is graphed in red, and the equation for the temperature with double-glazed windows is graphed in blue.

Is the red curve or the blue curve higher up on the grid at any given $x$ value?
Which does a better job of keeping the room warm: ordinary windows or double-glazed windows?

Click to see what happens to the room over a longer time period (15 hours).

Over this longer time period, is the red curve ever higher than the blue curve?
Approximately what will the eventual temperature of the room be if it has ordinary windows? degrees Celsius
Approximately what will the eventual temperature of the room be if it has double-glazed windows? degrees Celsius

Now the two graphs on the left show what occurs if the heat in the room is turned off while the temperature outside is $-5$ degrees Celsius. Again, the red curve ($y_1$) shows how the room cools off if it has ordinary windows, and the blue curve ($y_2$) shows how it cools off if it has double-glazed windows.

If the room has ordinary windows, approximately how long will it take for the temperature to drop below 10 degrees Celsius? hours
If the room has double-glazed windows, approximately how long will it take for the temperature to drop below 10 degrees Celsius? hours
When the temperature outside is $-5$ degrees Celsius, does the room cool off faster or slower than when the temperature outside is 5 degrees Celsius?
Click to see what happens to the room after 15 hours. If the temperature outside is $-5$ degrees Celsius, approximately what will the eventual temperature in the room be (whether the windows are double-glazed or not)? degrees Celsius