Math Is Fun Forum

English is my third language, after French and Arabic. When I joined the faculty of electrical and electronics engineering at AUB (in Beirut), my first two courses were in electronics. Their two scores were the lowest ones (around 70%, just enough to pass them) among my MS eight ones. This happened because, at the beginning, my English wasn't good enough to understand well their exam's problems. Later, my English became better, and my scores were around 95% for the remaining 6 courses (which were not related directly to electronics). The irony is that, after my studies, I became professional in electronics (for which I got the lowest scores) and not in the field of any of the six remaining courses (for which I got the highest scores).

Good remark.
In fact, in real life, queries/problems start (are given) always by words first.

It seems this is the case in almost all countries around the world.
I am afraid that a today's ordinary young person has to find ways to steal advanced scientific knowledge (unless his family is somehow rich), because, everywhere, ignorant people are easier to control than the well-educated ones.

So, during my school studies and high studies later, I wasn't looking for degrees but just good scientific knowledge, so that I could create my private business (which I did with an initial capital of what is equivalent to $100). So, after I passed well the MS eight courses (at the American University of Beirut, AUB) and my thesis was (still is) an innovative practical topology in communications (which I used in the 80's in my personal short-range RF links), I just returned home back to take care of my business (as a designer/producer in electronics).
But later I realized that I couldn't have the chance to do this if I was born in a mid-class family in what is known as the free world, because I would have no choice but to serve certain rich families (owning a certain big company related to electronics) by my knowledge.
For instance, at age 36, my father (who was brilliant in math) was hired to be the general manager of all branches in Middle East of a bank owned by certain French families. But about two years later, he was fired after he was a victim of a plot prepared by one of his older personnel to take his position. Unfortunately, his severe depression hit his two kidneys (in year 1959) and died soon after (I was 9). So, it became out of question for me to look for a job, for any position and any salary.

Being 75 now, I try my best to let my brain's cells be active by keeping them busy, as long as possible, in solving new things.
So, when I don't have a new design to think of (for the local market), I simply work on a new product which could be useful to me.
After all, my brain is the remaining organ in my living body that didn't reach its end of service

I wonder what the number of infected people (y) is supposed to be on the first day (x=1).
Is it y=1 for x=1?

y varies inversely with sqrt{x}.
But y = k/x means: y varies inversely with x, not sqrt(x), got it?

T varies jointly with the cube root of x and the square of d.
But T = cr{x}•sqrt{d}•k means: T varies jointly with the cube root of x and the square root of d, not square of d, got it?

Your time is short so you can't avoid misreading sometimes.

This is good news. You don't have time to feel bored

You are right, saying 0/0 is equal 1 always doesn't make sense, in math also in applied math (in real applications).
But we like it or not, it is well known that 'the boss is always right' and the boss's student is his teacher.

This is why I never argued with any of my teachers about what he taught me, even if I was very sure he is wrong about something.
I simply let my teachers hear what they like to hear while I saved every idea that looked right to me in my set of knowledge and rejected whatever sounded wrong/illogical (to me). And the ideas which I wasn't sure about, I kept them pending for further studies.
After all, I am afraid that, in life, a truth (in math or else) may not please everyone.

Sorry again, do I need searching the posts of your 'first' wrong solutions (not the ones later) when you used to apply x^2 + y^2 = r^2 instead of (x - h)^2 + (y - k)^2 = r^2?

Perfection is relative to one's need. In other words, one does something perfect when he can let it be as he needed to be. And, in this respect, you do very well, so far.

We may conclude that no human can reach the absolute perfection since a human has no reason to be interested 'really' in doing 'everything'.
For example, although my grades allowed me to join the faculty of medicine and be a doctor, I chose to be an engineer instead.

Kerim

It is right.

Please note that the following simple steps don't need knowing the hint.

In general, when we have two traces (here, a circle and a straight line) and we like to find at which point(s) they may intersect, we simply solve their two equations.

We have:
x^2 + y^2 = r^2
y = mx + b
And their unknown variables are x and y.

By substituding y, we get:
x^2 + (mx + b)^2 = r^2
x^2 + m^2*x^2 + 2*m*x*b + b^2 = r^2
(1 + m^2)*x^2 + 2*m*b*x + (b^2 - r^2) = 0

This is a quadratic equation in the form of:
A*x^2 + B*x + C =0
where
A = (1 + m^2)
B = 2*m*b
C = (b^2 - r^2)

Its solution is:
x1 = [-B - sqrt(delta)]/(2*A)
x2 = [-B + sqrt(delta)]/(2*A)

Where delta is:
B^2 - 4*A*C

Therefore the delta of our equation above is:
= B^2 - 4*A*C
= (2*m*b)^2 - 4*(1 + m^2)*(b^2 - r^2)
= 4*m^2*b^2 - 4*(b^2 - r^2 + m^2*b^2 - m^2*r^2)
= 4*(m^2*b^2 - b^2 + r^2 - m^2*b^2 + m^2*r^2)
= 4*(- b^2 + r^2 + m^2*r^2)
= 4*[r^2*(1 + m^2) - b^2]

Since the straight line intersects the circle at one point only, x1 and x2 of this equation should be equal.
They are equal if its delta=0, that is:
4*[r^2*(1 + m^2) - b^2] = 0
r^2*(1 + m^2) - b^2 = 0
r^2*(1 + m^2) = b^2