All Answers 358116 Answers

Nice answer Aziz....let's see how he might have gotten there......

Let x be the first number and y be the second  ...   so we have

x + y = 7.5

and

x * y = 9

Solving for y in the first equation, we have   y = 7.5 - x

And substitutuing for y in the second equation

x * (7.5 - x) = 9    simplifying, we have

-x^2 + 7.5 x   = 9      and rearranging, we have

x^2 - 7.5x + 9 = 0  subtract 9 from both sides

x^2 - 7.5x =  -9       now, take 1/2 of the coefficient on x (= 3.75), square it (=14.0625) , and add the square to both sides...so we have

x^2 - 7.5x + 14.0625 = 5.0625     factor the left side

(x - 3.75)^2  = 5.0625         take the square root of both sides

x - 3.75  = ± 2.25

So x = 3.75 - 2.25 = 1.5    or x = 3.75 + 2.25  = 6

And whatever we choose as x, y is the other value, so Aziz's answer of 1.5 and 6 is correct!!!!

We have the number of square feet x cost per square foot  = total cost......so.....

(15 x 11) x $2.65  =  $437.25

I assume we have

(√3 * (49)²) / 49    ????  ....if so, this gives us.....

49√3

lim as x → 6  of [√(x +3 ) - 3] / [ x - 6]

Note that this gives us the indeterminate form... ( 0 / 0 ).... when we take the limit !!!

When we have this situation, we can try to employ something called L'Hopital's Rule to find the limit.

Take the derivative of the numerator and the denominator.....this gives us

[(1/2)*(x + 3)^(-1/2)] / [1] = (1/2)*(x + 3)^(-1/2)

Now, take the limit of this as x → 6   so we have

[(1/2)*(6 + 3)^(-1/2) =  (1/2)*(9)^(-1/2) = (1/2)*(1/3) = 1/6

Note that the function doesn't actually exist at x = 6.......but, we don't care about that......we only care about what happens as we approach 6 from both sides.......!!!!!

This is pretty easy to solve and not much math is necessary.

Note that the pursuing cyclist "makes up" (30 - 18) = 12 km every hour.

And since he has to make up 19.2 km, we have

19.2 / 12 = 1.6 hr = 1 hr. 36 min.

Thank you David.

You have expressed is a wonderful show of appreciation.

I really appreciate it :)

It is like swimming in an ocean of near infinite complexity. Even so, I do recognize the “water,” such as the plus and minus signs.

I suspect you do too, and much more.

~~D~~

(3 + 10) = 3 + 10

DavidQD

From my brief knowledge about the proof of Fermat's Last Theorem....I don't think that most math histrorians believe that Fermat had enough math available to him in his era to prove his own conjecture. I believe - if I remember correctly - that Wiles brought two seemingly unrelated areas of math together (ellliptical curves and something else??) to establish a proof.

I once asked  one of my math professors if he could explain Wiles' proof.....he told me that it was WAY above his pay grade.....I knew then that it must be pretty difficult to understand !!!!

Thanks Melody.

This was an attempt to demonstrate that because an integer squared (x^2) (where all such numbers have an odd number of positive divisors) is equal to another number squared (y^2), plus an additional factor (5) that the value of (x) and/or (y) cannot be integers. This means it is equal to a number that has an even number of divisors, or it does not have an even number of divisors to begin with. Because of this contradiction, the number cannot be an integer. Extending on this was an attempt to explain why an irrational number had characteristics of both even and odd numbers.

The explanation I presented is incomplete. This has been rolling around in my head for years, and it seemed functional, until an examination in a proper light. On closer inspection, it seems it is an incomplete amalgamation of proofs by contradiction and a hint of a proof by infinite decent thrown in for good measure (Fermat would be appalled). Mostly, it is a succession of incongruous comparisons that are not made compatible mathematically. Very lame. This is worse than mixing up metaphors. So, thank you Melody, I now see the darkness at the end of the tunnel.

 Your proof is to the point clear. CPhill’s, detailed and elegant, proof clearly demonstrates the paradox of (even and odd) parity.

Somewhere, there is a direct proof (instead of proofs by contradiction) of irrational numbers. Probably the best place to look is in Andrew John Wiles’ proof of Fermat’s Last Theorem. These maths are far outside of my skill set, and always will be. Even so, after dozens of readings, I have gleaned insights from them.

To find a direct proof for irrational numbers could lead to maths that close the gaps on Cantor’s Alephs of infinites. Considering there are more irrational numbers than integers, the value of this proof may be enormous. Adding to this, Wiles’ proof includes proofs of the Modularity Theorem for semistable elliptic curves, with these kinds of proofs, all three of Georg Cantor’s infinites may converge. Wiles’ proof seems like an excellent place to start.

Noting the post on Fermat, there is one question that remains about Fermat’s Last Theorem: Did Fermat really have the remarkable proof he wrote of in the margin of his book? Considering it took advanced, late twentieth century mathematics to solve it, I seriously doubt it. However, it is an intriguing thought that seventeenth century math may hold a solution.

This forum is great. The mathematicians who visit and especially the ones who live here are truly extraordinary.

Thank you Melody and CPhill!

~~D~~

...................................................................................................................................................

CPhill, is it correct?

1.5 & 6.

1.5 + 6 = 7.5

1.5*6 = 9.

The answer is a book! I got the answer!

I posted the answer and I am waiting.

WOW Honga,

This is really impressive.

To be truthful I spend so much time on administrating this site that i do not always look at answers that I am sure I would find greatly illuminating.  

I am very glad that I looked at this answer.  Your explanation is brilliant.

I hope that you stick around to WOW us a lot more.

Thank you.

There are one billion ( 10⁹ ) nanometers in a meter

And there are  1609.344 meters in a mile.

And the diameter of each gold atom is 2 x .135 = .270nm

So we have.....

(1609.344 x 10⁹ ) nm /  (.270) nm = about 5,960,533,333,333   gold atoms...... (about 6 trillion)....!!!!

Do you mean the mean? 

You can use the on-site calculator to solve this.

Simply distribute the -3 and 2 first.

Then, collect "like" terms and that's it! For example, 2x + x = 3x & -2v + 3v = 1v = v.

You should be able to do this now!

-27 = -6 - p

First, add 6 to both sides since you want to solve for p!

-21 = -p.

Note that we have -p and not p. So, we have to divide both sides by -1 giving us our final answer:

p = 21.

This is the other answer. The answer that AzizHusain gave is an improper fraction.

for the mixed number, the following is the answer.

$$2\frac{2}{5}$$ That is the answer.

s = d/t

s = 300*10^5 / 3.5*10^8

Now, we can simply cancel out 10^5 and convert 10^8 to 10^3 since 8-5 = 3. Note that 10^3 is in the denominator meaning we move the decimal place 3 times to the left. 

Also, we can just normally divide 300 by 3.5 (same as 7/2): 300 * 2 / 7 = 600/7. Since 600/7 is in the numerator, we can take its decimal and move the decimal place 3 times to the left.

So, we end up with 600/7 * 10^-3 unit/unit [m/s for example] = ~0.0857 unit/unit.

Think of 2.4 as 24. How do we get from 2.4 to 24?? 

Notice that you have to move the decimal place to the right once. This is the same as multiplying by 10. So, if we want to get 2.4, we have to divide 24 by 10!

24/10 = 2.4

Now, we just have to simplify 24/10 by dividing both the numerator and denominator by 2:

12/5.

Nice BIG explanation, ND !!!   REALLY BIG !!!

I want to point out someting...."FOIL" is a useful method for expanding multiplied binomials. We could also do this....

(m - 50) (m + 18)

"Distribute" the "m" in the first set of parenthesis over the two terms in the other set of parenthesis...so we have......

m * m  + m *18   =  m² + 18m

Them, do the same thing with the "-50"    ...and we have......

-50m - 900

And combining like terms, we have

m² + 18m - 50 m - 900  =

m² - 32m - 900   ......just as NinjaDevo found

The only reason I mention this is that "FOIL"  can't be effectively used if we are multiplying polynomials that involve more than two terms. In that case, another "ditribution" method is often necessary......!!!!