All of us have probably been exposed to questions such as: "What are the applications of group theory...".
This is not the subject of this MO question.

Here is a little newspaper article that I found inspiring:


  Madam, – In response to Marc Morgan’s question, “Does mathematics have any practical value?” (November 8th), I wish to respond as follows.
  
  Apart from its direct applications to electrical circuits and machinery, electronics (including circuit design and computer hardware), computer software (including cryptography for internet transaction security, business software, anti-virus software and games), telephones, mobile phones, fax machines, radio and television broadcasting systems, antenna design, computer game consoles, hand-held devices such as iPods, architecture and construction, automobile design and fabrication, space travel, GPS systems, radar, X-ray machines, medical scanners, particle research, meteorology, satellites, all of physics and much of chemistry, the answer is probably “No”. – Yours, etc,
  
  PAUL DUNNE,
  The Irish Times - Wednesday, November 10, 2010



The above article article seems to provide an ideal source of solutions to a perennial problem: How to tell something interesting about math to non-mathematicians, without losing your audience?
However, I am embarrassed to admit that I have no idea what kind of math gets used for antenna designs, computer game consoles, GPS systems, etc.

I would like to have a list applications of math,
from the point of view of the applications.

To make sure that each answer is sufficiently structured and developed, I shall impose some restrictions on their format. Each answer should contain the following three parts, roughly of the same size:

• Start by the description of a practical problem that any layman can understand.
• Then I would appreciate to have an explanation of why it is difficult to solve without mathematical tools.
• Finally, there should be a little explanation of the kind of math that gets used in the solution.

♦♦  My ultimate goal is to have a nice collection of examples of applications of mathematics, for the purpose of casual discussions with non-mathematicians. ♦♦

As usual with community-wiki questions: one answer per post.

Sending a man to the Moon (and back).

Hilbert once remarked half-jokingly that catching a fly on the Moon would be the most important technological achievement. "Why? "Because the auxiliary technical problems which would have to be solved for such a result to be achieved imply the solution of almost all the material difficulties of mankind." (Quoted from Hilbert-Courant by Constance Reid, Springer, 1986, p. 92). 

The task obviously required solving plenty of scientific and technological problems. But the key breakthrough that made it all possible was Richard Arenstorf's  discovery of a stable 8-shaped orbit between the Earth and the Moon. This involved the development of a numerical algorithm for solving the restricted three-body problem which is just a special non-linear second order ODE  (see also my answer to the previous MO question).  

Another orbit, also mapped by Arenstorf, was later used in the dramatic rescue of the Apollo 13 crew. 
