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Hello I am having some difficulty with this problem.
Two positive numbers differ by 11, and their square roots differ by 1. Find
the numbers.
Difference between the numbers is 11.
This means (x-y)=11
substituting in our main equation, we get:
11*(x+y)=1
(x+y)=0.5
Now we arrive at two simple linear equations:
(x+y)=0.5
(x-y)=11
solve for 'x' and 'y' from the above eqns to get:
x=5.75, y=-5.25
$\displaystyleQuote:
Originally Posted by pashah
x-y=11
$
$\displaystyle
\sqrt{x}-\sqrt{y}=1
$
Trial and error indicates 36 and 25 are suitable numbers.
RonL
Really fast reply. Much appreciated.
A more elegant solution.Quote:
Originally Posted by pashah
You have,
$\displaystyle x-y=11$
Use diffrence of two squares,
$\displaystyle (\sqrt{x}-\sqrt{y})(\sqrt{x}+\sqrt{y})=11$
But,
$\displaystyle \sqrt{x}-\sqrt{y}=1$
Thus,
$\displaystyle \sqrt{x}+\sqrt{y}=11$
Therefore, the two equation you have are,
$\displaystyle \sqrt{x}+\sqrt{y}=11$
$\displaystyle \sqrt{x}-\sqrt{y}=1$
Add then subtract them,
$\displaystyle 2\sqrt{x}=11\rightarrow x=36$
$\displaystyle 2\sqrt{y}=10\rightarrow x=25$
Thank you for the expert help. One thing you did forget to mention about the four individuals. They were all megalomaniacs who were hell bent on conquering the known world. Unlike yourself I don't see where they were interested in helping others overcome difficulties. Thanks Again.
I would place Hannibal of Carthage on the list since he is my favorite (ever I seen Hannibal vs. Rome on Histroy Channel Special) but he never did try to take over the world.Quote:
Originally Posted by pashah
That's a strange coincidence since I am also fond of Hannibal. His father Hamilcar was quite an impressive leader as well. Although, I must say they seem to have commited some horrible atrocities in their conquests. Hannibal was known to slaughter entire companies of his own men for fear that they would fall prey to the Roman legions of Scipio. I suspect his biggest mistake was limiting his tactics. He was redundant and consequently an observant Scipio would later adopt his tactics and use those very same tactics to defeat him in a decisive battle.
Did you know that Hannibal was originally from an ancient tribe of Spanish descent?
Won spectacular victories in battle, lost war due to inability toQuote:
Originally Posted by ThePerfectHacker
overcome political constraints.
Compare with oneone like W S Churchill - dreadfull when interfereing
with the running of campaigns but understood the (geo-) political
side better than the enemies of the UK - result: the UK on winning side
in one of the most important conflicts in its history.
RonL
Hello, pashah!
TPHacker's solution is elegant.
It still can be solve by "normal" methods.
Quote:
Two positive numbers differ by 11, and their square roots differ by 1.
Find the numbers.
We have: .$\displaystyle \begin{array}{cc}(1)\;\;x \:- \:y\;=\;\;11 \\ (2)\;\sqrt{x} - \sqrt{y}\:=\:1\end{array}$
From (2), we have: .$\displaystyle \sqrt{y} \,= \,\sqrt{x}-1\quad\Rightarrow\quad y \,= \,(\sqrt{x} - 1)^2$
Substitute into (1): .$\displaystyle x - (\sqrt{x} - 1)^2\:=\:11\quad\Rightarrow\quad x - (x - 2\sqrt{x} + 1) \:= \:11$
Hence: .$\displaystyle 2\sqrt{x} - 1 \:=\:11\quad\Rightarrow\quad 2\sqrt{x}\,=\,$$\displaystyle 12\quad\Rightarrow\quad \sqrt{x}\,=\,6\quad\Rightarrow\quad x = 36$
Therefore: .$\displaystyle \boxed{x = 36,\;y = 25}$
