MHB Comparing $7$ and $\sqrt{2}+\sqrt{5}+\sqrt{11}$: Which is Bigger?

[Albert1]

compare :
$7$, and $\sqrt 2+\sqrt 5 + \sqrt {11}$
which one is bigger?

 

[Greg]

Spoiler

Approximating the roots to 3 significant digits, we have$$\sqrt2+\sqrt5+\sqrt{11}\approx1.41+2.23+3.31=6.95,6.95+3\cdot0.009=6.977\lt7$$

 

[Albert1]

Spoiler

Approximating the roots to 3 significant digits, we have$$\sqrt2+\sqrt5+\sqrt{11}\approx1.41+2.23+3.31=6.95,6.95+3\cdot0.009=6.977\lt7$$

now ,try to prove it,without using calculator

 

[MarkFL]

now ,try to prove it,without using calculator

It is fairly easy to compute square roots by hand to two decimal places...[cs=Spiteful]spiteful.gif[/cs]

 

[Albert1]

It is fairly easy to compute square roots by hand to two decimal places...[cs=Spiteful]spiteful.gif[/cs]

you don't have to compute the square roots of those numbers ,use some tricks

 

[anemone]

I am going to use a trick someone taught me from this forum to solve for this challenge:

Spoiler

Note that

$288<289\,\,\implies2(12^2)<17^2$ or $(\sqrt{2}<\dfrac{17}{12})---(1)$

$80<81\,\,\implies5(4^2)<9^2$ or $(\sqrt{5}<\dfrac{9}{4})---(2)$

$99<100\,\,\implies11(3^2)<10^2$ or $(\sqrt{11}<\dfrac{10}{3})---(3)$

Adding the inequalities in (2), (2) and (3) up gives us the answer:

$\sqrt{2}+\sqrt{5}+\sqrt{11}<\dfrac{17}{12}+\dfrac{9}{4}+\dfrac{10}{3}=7$

 

[Albert1]

I am going to use a trick someone taught me from this forum to solve for this challenge:

Spoiler

Note that

$288<289\,\,\implies2(12^2)<17^2$ or $(\sqrt{2}<\dfrac{17}{12})---(1)$

$80<81\,\,\implies5(4^2)<9^2$ or $(\sqrt{5}<\dfrac{9}{4})---(2)$

$99<100\,\,\implies11(3^2)<10^2$ or $(\sqrt{11}<\dfrac{10}{3})---(3)$

Adding the inequalities in (1), (2) and (3) up gives us the answer:

$\sqrt{2}+\sqrt{5}+\sqrt{11}<\dfrac{17}{12}+\dfrac{9}{4}+\dfrac{10}{3}=7$

very smart!

 

[Fernando Revilla]

Another way.

Spoiler

We have the following equivalences
$$\begin{aligned}\sqrt{2}+\sqrt{5}+\sqrt{11}<7&\Leftrightarrow \sqrt{5}+\sqrt{11}<7-\sqrt{2}\\
&\Leftrightarrow\left(\sqrt{5}+\sqrt{11}\right)^2<\left(7-\sqrt{2}\right)^2 \\&\Leftrightarrow 16+2\sqrt{55}<51-14\sqrt{2}\\
&\Leftrightarrow 2\sqrt{55}+14\sqrt{2}<35\\&\Leftrightarrow \left(2\sqrt{55}+14\sqrt{2}\right)^2<35^2\\&\Leftrightarrow 612+56\sqrt{110}<1225\\
&\Leftrightarrow 56\sqrt{110}<613\\
&\Leftrightarrow \left(56\sqrt{110}\right)^2<613^2\\
&\Leftrightarrow 344960<375769.\end{aligned}$$