How fast will my coffee get cold?

With a not so difficult mathematical formula you can calculate exactly at what time will your cup of coffee (or tee) get cold (or to a certain temperature). It all depends on the room temperature, if it's colder the coffee will get colder faster and if it's a hot summer's day you coffee will stay warm for longer.

Let's take a look at the formula:







This is called a first-order differential equation. The T refers to the temperature at any given time t and A is the room temperature. The constant k indicates the rate at which the temperature will change.

Let's solve this equation:


Now we need to A (room temperature), B (which we can find by knowing the initial temperature of the coffee) and k (which is a bit more difficult to find).

Let's say A = 22˚C and that since water boils at 100˚C and we immediately make our coffee, let's assume that T(0) = 90˚C (Taking into account that it will cool off a bit when pouring and stirring).

Now we have the following:

To find the value of k we need to do some experimentation, where we determine the temperature of the coffee after some specific time. So let's say that after t=2 minutes we find that the temperature of the coffee is now 85˚C (I'm just guessing a value, for exact results one would have to take the actual temperature of the coffee after two minutes).

So now we find k:

Hence the equation becomes:

If I now want to know the temperature of my coffee after 20 minutes, let t = 20, then

and you will see that as time goes by the temperature will drop at a slower rate until it reaches the room temperature.

Happy coffee-drinking!

Chocolate math

Love math - Put on your positive thinking hat

I love math. Who doesn't?
If you don't it must be because you haven't met with its amazing properties and applications.

For many people it's understanding math that is the problem and from that the negativity spirals out onto those around them and hence the general dislike of mathematics.

I believe that everyone are able to do math and understand the basic ideas, excluding perhaps all the complicated concepts one might encounter past a first year university level. The problem is the attitude going in, the believe that you cannot do something before you even tried. With that kind of thinking no one will ever get anything done.

Let's all put on own positive thinking hats wherever we go in this world and not only embrace math, but all seemingly difficult tasks that are throw unto our paths everyday. The next time someone wants to teach you something about math, say yes and you might just be surprised.

Our minds can store an endless amount of information so why not use some of that space to learn something new and interesting every day?

Happy positive thinking!

How not to expand an equation

The correct answer would be:


and we can make use of the Binomial theorem to find all the other terms.

Trigonometry Series (Part 4)

Today we look at several formulas which tend to be helpful when we are solving trigonometric equations or working with specific angles. They are also useful for simplifying expressions.

Addition formulas:


The formula for tan can be derived by making use of the formulas for sin and cos.


Subtraction formulas:

 
The subtraction formulas can be easily derived from the addition formulas by making use of the angle identities in Part 3.

Double-angle formulas:


These formulas can be derived from the addition formulas by letting y=x and using some identities.

Half-angle formulas:



These formulas can be obtained by rewriting the double-angle formulas.

Next time we will consider a few problems and examples, showing how we can apply and use all the information that we have from the previous parts in the series.