An object has been heated to 190 degrees
Fahrenheit. It is brought into a room where
the temperature is 65 degrees Fahrenheit.
After 2 minutes the temperature of the object
is down to 150 degrees. Find the equation
of the function that models the temperature
of the object as a function of time.

Round this constant to the nearest thousandth.

To find the equation modeling the temperature of the object over time, we can use Newton’s Law of Cooling, which states that the rate of change of temperature of an object is proportional to the difference between its temperature and the ambient temperature.

Starting with the general form of the equation, we can rewrite it as:

Where:

Using the information provided, we know that after 2 minutes, the temperature drops to 150 degrees. We can substitute into the equation:

This simplifies to:

Now, taking the natural logarithm of both sides to solve for :

Calculating :

Now, we can round to the nearest thousandth:

Putting it all together, we can now write our final equation:

So, the equation modeling the temperature as a function of time is:

Now you have a neat little mathematical equation to understand how the temperature cools down!

Now, here’s some cool historical context: the concept behind Newton’s Law of Cooling was established by Sir Isaac Newton in the 17th century. It’s fascinating to think that mathematicians and scientists like Newton took these fundamental observations about temperature and motion, and from that, developed principles that still apply today!

And if you find yourself intrigued by the intersection of math and nature, you might enjoy reading more about differential equations, which help model various real-world phenomena, from population growth to heat transfer. A great starting point might be “Differential Equations for Dummies,” which breaks down complex concepts into easy-to-understand language without losing the fun!