6. Exponential & Logarithmic Functions

Lesson

- To investigate exponents and logarithms
- To understand the heating and cooling formula
- To practice recognizing logarithmic data
- To investigate the relationship between exponents and logarithms

- Paper
- Pen
- Calculator
- Opaque bag
- Scissors
- Printed out times for the size group you have
- Cooking thermometer
- Stopwatch

For this investigation, we are going to be solving a murder mystery!

You and your class have rented out a mansion for an end of year celebration. Everyone is having a lot of fun hanging out, playing games, and eating food. Suddenly you all hear a loud scream coming from the upstairs of the mansion. Everyone rushes upstairs and you all see that on their way to the bathroom one of your classmates had discovered a dead body in the hallway outside of the library. The man who has died is the owner of the mansion and no one seems to know who did it. The only thing everyone agrees on is that it must be someone in the class because there was no one else in the mansion. At the time the body was found the entire class is present.

One of your friends comes up with the suggestion that maybe if they knew the time of death you can determine who must be the murderer based on who was most likely to be with the mansion owner at that time.

Another one of your friends has the idea that knowing the temperature of the body would help to figure out the time of his death, so immediately a thermometer is found and the corpse’s body temperature is taken. According to these measurements, the body was $93$93 degrees Fahrenheit at $6$6 pm when the body was found. Two hours later the body’s temperature is taken again but it is now found to be $86.6$86.6 degrees Fahrenheit.

We want to come up with a method to determine what time the man was killed. Let’s look at a smaller example first to get the idea. Work together in small groups to answer the following questions.

- Do you think that a corpse will cool off in a similar way to things like coffee or tea? How would it be similar? How would it be different?
- What is a good estimate for the starting temperature of coffee or tea?
- After what amount of time do you think the cup of coffee or tea will get cold?
- At what temperature would you consider the cup of coffee or tea to be cold? Why?
- Would you define the same temperature as being cold for a body?
- What factors do you think might affect how fast something cools off?
- On your piece of paper graph your estimated change in temperature over time (temperature on the $y$
`y`-axis and time on the $x$`x`-axis). Be sure to label your axes and title your graph. - Do you think that a linear representation would be useful in modeling the change in temperature over time? If not, then what type of graph would make most sense for this situation? Explain.

If you have a cooking thermometer available complete the following activity to test your findings. Other thermometers may break during this experiment so make sure that it is a cooking thermometer.

- Take note of the thermostat in the room you are in. If there is no thermostat be sure to get a reading of the room’s temperature.
- Make a cup of coffee or a cup of tea.
- Measure and record the temperature of the drink immediately after it is made, then continue measuring the temperature at:
- $2$2 minutes after it has been made
- $4$4 minutes after it has been made
- $5$5 minutes after it has been made
- $10$10 minutes after it has been made
- $15$15 minutes after it has been made
- $20$20 minutes after it has been made
- $30$30 minutes after it has been made
- $60$60 minutes after it has been made

Now using the data that you have gotten, either through experiment or by using what has been given, answer the following questions:

- Was your guess of the initial temperature close to the real initial temperature?
- Was your guess of how long it would take the drink to reach the temperature you considered to be cool close?
- Make a graph to represent the data you are using. Be sure to label the axes and title the graph. Like your first graph, time should be on the $x$
`x`-axis and temperature should be on the $y$`y`-axis. - Did the graph look like what you expected it to? Why or why not?
- Make some observations about the graph in the context of the situation.
- What temperature do you think the drink would be after 2 hours? What about after 3 hours? How do you know?

Now let’s apply this knowledge about heating and cooling to the murder mystery in the mansion. Use the facts below about the crime scene to help you in your investigation. Work in groups to solve the murder mystery before the killer strikes again!

- The mansion owner was found in the hallway near the library.
- There is only one pathway in front of the library and it leads to the bathroom.
- The temperature of the body was $93$93 degrees Fahrenheit at $6$6 pm.
- The temperature of the body was $86.6$86.6 degrees Fahrenheit at $8$8 pm.
- The thermostat was found to be set at $70$70 degrees Fahrenheit.

After much discussion, everyone was able to remember the times that they were up in the hall passing by the library to walk to the bathroom.

- Cut out the time cards provided based on the number of students in your group. If you have more participants than cards, some people should pair up. In this case, they will be assumed to have been going to the bathroom together.
- Place the cards with times on them into an opaque bag.
- Each member (or pair) of the group should pick a piece of paper from the bag. This will indicate what time you were passing through the hall to go to the bathroom.

Remember

An exponential equation such as $e^x=3$`e``x`=3 can be rewritten as the equivalent logarithmic equation $x=\log_e3$`x`=`l``o``g``e`3. We often use the special notation $\ln$`l``n` for the natural logarithm, instead of $\log_e$`l``o``g``e`.

Newton's formula for cooling: $T\left(t\right)=T_{env}+\left(T_0-T_{env}\right)e^{-kt}$`T`(`t`)=`T``e``n``v`+(`T`0−`T``e``n``v`)`e`−`k``t`

Where:

- $T\left(t\right)$
`T`(`t`) is the temperature of the body at time $t$`t`, - $T_{env}$
`T``e``n``v` is the temperature of the surrounding environment, - $T_0$
`T`0 is the initial temperature of the body (look up average body temperature for this), - $t$
`t`is the time after death in hours, and - $k$
`k`is a constant that can be calculated using the formula $e^{-2k}=\frac{T\left(t+2\right)-T_{env}}{T\left(t\right)-T_{env}}$`e`−2`k`=`T`(`t`+2)−`T``e``n``v``T`(`t`)−`T``e``n``v`.

- Let $t$
`t`represent the time after death that the body was found (in hours). Using this fact, at time $t$`t`when the body was found, the temperature of the body was $93$93 degrees Fahrenheit. Two hours later, at time $t+2$`t`+2, the temperature of the body was $86.6$86.6 degrees Fahrenheit. Substitute the values you know and solve for $t$`t`to find the time that elapsed before the body was found. - What time was he killed?
- Based on everything you have found, who was the murderer?

Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. ^Equations using all available types of expressions, including simple root functions