Math Is Fun Forum

For instance, when I was a student in schools then universities, I used to (actually I had to) follow, in every exam, the definitions that pleased the teacher in charge of it.
After graduation, I was able to start a private business as a designer in electronics for the local consumers (I am 74). And since then, I used to follow at work what pleases me only

3^b = 9√3
3^b = 3^2 * 3^1/2
3^b = 3^(2+1/2)
3^b = 3^(5/2)

b = 5/2

It seems no one around here recall well the first definition of parallelism he heard of at school It was:
Two straight lines are said parallel if they don’t intersect.
Kids see this wrong definition as being true because they see the geometrical figures on their sheets of paper only or on any other planes.

Only when these kids will grow up and start to see them (geometrical figures) in space, they will know it is wrong and it needs to be updated as:
Two straight lines are said parallel if they don’t intersect and are on the same plane.

As long they will not need to learn the perspective geometry, the second definition is all what they need to know about how parallelism is defined.

But those who will need learning it (the perspective geometry), they will hear the complete definition of parallelism which is:
Two straight lines are said parallel if they do intersect at infinity.

I guess, it is out of question introducing parallelism to kids using its third and best definition, right?

Similarly, introducing the fact of Creation to the kids of humanity, our very old ancestors, by unreal tales was very important at those early times, as the first definition of parallelism is also wrong but important for the kids at elementary school.

So now, I don't need to hear any tale to know that there is a supernatural Will/Power that let me (actually forced me to) exist temporarily in a realm which is defined/limited by time and space. Fortunately, like any other human, I was also given, by design, a human brain in which I used to trust its power to no limit (equivalent to trusting its maker to no limit). It helped me discover, with time, the logical answers to all important/crucial questions, related to my own existence and the real world on earth, which I was looking for.

Kerim

By the way, we get the same result (Vb) if:

(Va) = 12600 liters (was 16,800 liters)
AB = 360 miles (was 420 miles)
(Vt) = 4200 liters (as before)
(Ct) = 10 liters/mile (as before)

Perhaps you mean as:
cube root y * cube root y * cube root y = cube root y^3

For example:
cube root 27 * cube root 27 * cube root 27 = cube root 19683 (which = 27)

Thank you for introducing this method which I am not familiar to.

Its crucial step is to find ‘p’ the inverse of Z (modulo n) which leads us to solve another equation with two unknowns as:
Z*p = n*q + 1 , where ‘p’ and ‘q’ are also natural numbers.

In this exercise it is:
12*p = 17*q + 1

And it is clearly simpler than the original one:
17*A + 29*B = 1000

I wonder if this will be the case always.

Kerim

It is always good to know how to use a math tool when it is available.
On my side, I see myself fortunate that I can still use Excel (a very old version) to solve various mathematical problems related to my designs (in electronics).

It happens that, since very long, I have no more the privilege, as most engineers in the world have, to download a tool/program, free or paid, for advanced math (besides other CAD) to save time in solving complex math problems. But, to me in the least, this is not bad at all, because I enjoy, since I was young (74-year now) playing with numbers, parameters, variables, formulas and equations. I also used, since after graduation, to write CPU then MCU codes in assembly language only for my various designed controllers (needed by the local consumers in every period of time) because I am also not allowed to download any high language compiler.
Anyway, being a rather old man, I expect losing, gradually with time, the last helper I still have, my human brain

About math tools, I recall I purchased a programmable TI calculator while studying for MS degree. It could be programmed to solve non-linear equations. At a final exam, I entered the non-linear equation, somehow complex, related to the given problem and let the calculator find out the answer. The professor didn't expect that a student could find the answer in the limited exam's time. He told me that what I did is like cheating. So, I gave him my calculator and ask him to use it, as I did, and have the answer.
He replied: "But I don't know how to use it". I went on saying: "Me too, I had no idea how to use it. But I spent enough time to learn its various functions to help me save time when necessary. This calculator cannot find out the equation of the problem. It can solve it only". He apologized and gave me the highest grade.

I will refer here to your original diagram (I guess you still have it).

I used to tell my students that, in general, finding how to solve a problem has two main directions, forwards and backwards.
To solve this exercise, walking backwards makes it a simple one.
   
Let us assume first that the full range (R) of the missile was used.
In this case, the warship had to be on the circle whose center is ‘Y’ and radius is (R).
If we draw this circle, it becomes clear that WX is the shortest distance from W to this circle.
Therefore, the mathematical solution of T=f(D,R,S1,S2,S3) could take advantage of Pythagoras theorem of the right triangle SWY.

One may think that it is not necessarily to use the full range of the missile, say (k*R) instead of (R), where 0<k<1.
The same reasoning above also applies in case (k*R). Only the radius of the circle becomes (k*R) instead of (R).
This means that T_new = f(D,k*R,S1,S2,S3). That is (R) is replaced with (k*R) in the final formula of (T).
And by analyzing the variation of the function (T_new) with the variable parameter (k), we will find out that T_new > T for all values of (k).