# Proposed solution 2

- Author:
- Bea Kristinsdóttir

## Suspect

Did the language teacher, Nicci Vulture, stab the mathematics teacher from behind with a papercutter …? Is that even possible? With a nagging feeling that something is not completely right, you slowly
walk back into the school building and to the principal’s office to report your conclusions
to your chief. It is a little bit warmer inside than outside and a shiver runs
down your back. You take a few deep breaths and try to calm down. Your conclusions
are surely on target…

## Aha!

On your way to the principal’s office, it suddenly strikes you. The temperature outdoors
is falling! At 9 o’clock it was 25°C, at 10 o’clock it was just 22°C, and now—at
almost 11 o’clock—the temperature is down to about 20.5°C. Your model is wrong,
but how do you find a model that takes into account an outdoor temperature that is
falling? You will have to solve the basic differential equation again. Of course, it
wasn’t the language teacher Nicci Vulture, 77 years old, who killed the mathematics
teacher, you think while you are finishing off your new calculations. It must have
been … You search in the list of the suspects. How strange… Who did kill the mathematics
teacher?

## Let us continue...

The temperature is falling at 3°C per hour since at least 8:30 that morning. Newton’s
cooling law claims that the speed of cooling is proportional to the difference in temperature.
You therefore set up the differential equation

`T′ = k · (T – (22 – 3t))`

, where
the time t once again is measured in hours after 10:00. Open up the GeoGebra CAS
window from the menu **Show > CAS**and type`SolveODE[y′ = k(y – (22 – 3x))]`

_{1}k e^{kx}– 3kx + 22k – 3)/k = c_{1}e^{kx}– 3x + 22 – 3/k. Now you create a new model function`m2(x) = c e^(k x) – 3x + 22 – 3/k`

`Fit[{A,B}, m2]`