All Answers 358149 Answers

$${\sqrt{{\mathtt{2\,116}}}} = {\mathtt{46}}$$

I get a similar(ish) expression to Bertie, except that I count from the first person not to be eliminated on the first rotation round the circle.

$$p=2(n-2^{floor(\log_2{n})})+1$$

This generates the following positions, p, for n from 1 to 16

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$$0.5=\frac{5}{10}=\frac{1}{2}$$

@@ End of Day Wrap   Wed 3/6/15   Sydney, Australia Time   8:10pm   ♪ ♫

Hi everyone, it is great to be posting on the correct side of midnight for a change 

Today our fabulous answers were provided by Civonamzuk, Alan, Mathcad, Radix, thejamesmachine, NinjaDevo, DarkBlaze347, LilSavvyBih, Radix and Bertie.     Thanks all.   

Thurs 4/6/15

If you would like to comment on other site issues please do so on the Lantern Thread.  Thank you.    

Interest Posts: 

FTJ means 'For the juniors' 

Thanks Bertie,

The penny just dropped, at least I understand what it being asked now.     

$$\\\sqrt{\frac{1}{x}}=3\\\\
\frac{1}{x}=9\\\\
\frac{1}{x}=\frac{9}{1}\\\\
\frac{x}{1}=\frac{1}{9}\\\\
x=\frac{1}{9}\\\\$$

16.7834 degrees

$${\mathtt{0.783\: \!4}}{\mathtt{\,\times\,}}{\mathtt{60}} = {\mathtt{47.004}}$$

$${\mathtt{0.004}}{\mathtt{\,\times\,}}{\mathtt{60}} = {\frac{{\mathtt{6}}}{{\mathtt{25}}}} = {\mathtt{0.24}}$$

so

16.7834 degrees = 16 degrees 47 minutes and 0.24 seconds

Assuming that the events are independent then

$$\\\frac{1}{3}\times P(Cat)=\frac{2}{15}\\\\
P(Cat)=\frac{2}{15}\div \frac{1}{3} \\\\
P(Cat)=\frac{2}{15}\times \frac{3}{1} \\\\
P(Cat)=\frac{2}{\not{15}^5}\times \frac{\not{3}^1}{1} \\\\
P(Cat)=\frac{2}{5} \\\\$$

Thanks for that great answer NinjaDevo  

That is a great answer  MathsGod1     

$$\\865.28=800(1+r)^2\\\\
865.28/800=(1+r)^2\\\\
\sqrt{865.28/800}=1+r\\\\
\sqrt{865.28/800}-1=r\\\\$$

$${\sqrt{{\frac{{\mathtt{865.28}}}{{\mathtt{800}}}}}}{\mathtt{\,-\,}}{\mathtt{1}} = {\frac{{\mathtt{1}}}{{\mathtt{25}}}} = {\mathtt{0.04}}$$    

so the rate is 4% per annum

Amount after 3 years

FV= 800(1.04)^3

$${\mathtt{800}}{\mathtt{\,\times\,}}{\left({\mathtt{1.04}}\right)}^{{\mathtt{3}}} = {\mathtt{899.891\: \!2}}$$

The compound interest after 3 years will be  $99.89

$${\mathtt{899.89}}{\mathtt{\,-\,}}{\mathtt{800}} = {\frac{{\mathtt{9\,989}}}{{\mathtt{100}}}} = {\mathtt{99.89}}$$       

The difference is 9 each time so part of it will be 9n

the first one is 9*1-6

the second is 9*2-6    etc

so

Tn=9n-6

( this is the same as Alan's answer)

Thejamesmachine, you need to show how you get the answers :/

As yet I don't know how to prove this, but a formula that seems to work is

$$\displaystyle p=2\left(n-2^{\bmod\left(\frac{(\log(n)-0.001)}{\log(2)}\right)\right)-1.$$

The logs are to base 10 and mod (I hope, otherwise I need to find another operator,) delivers the integer part of the expression within the brackets.

p is measured from the first person to be eliminated, being the number of positions further on around the circle.

For example n = 6 will return a value of p = 3. So, from the first person eliminated count to the third position further round the circle.

$$\\3/4=0.75\\
0.75X=15\\
$divide both sides by 0.74$\\
X=15/0.75$$

Looks like a simple arithmetic progression with each term 9 more than the previous one. The n'th term could be written as:  3+9*(n-1)  (where n≥1)

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For one example do a search on "surveying"

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The exponent is 5/8 MG not 5/6.

$${{\mathtt{8}}}^{\left({\frac{{\mathtt{5}}}{{\mathtt{8}}}}\right)} = {\mathtt{3.668\: \!016\: \!172\: \!818\: \!684\: \!9}}$$

She was given 2 boards at 126 inches ND, so total length she's received is 2*10.5 + 8 = 29 feet.  Still not enough for the sheep pen!

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You can use the quadratic formula to solve this; or just use the solver here:

$${{\mathtt{x}}}^{{\mathtt{2}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{0.17}}{\mathtt{\,\times\,}}{\mathtt{x}}{\mathtt{\,-\,}}{\mathtt{1.81}}{\mathtt{\,\times\,}}{{\mathtt{10}}}^{-{\mathtt{4}}} = {\mathtt{0}} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = {\mathtt{\,-\,}}{\frac{\left({\mathtt{23}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{14}}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{85}}\right)}{{\mathtt{1\,000}}}}\\
{\mathtt{x}} = {\frac{\left({\mathtt{23}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{14}}}}{\mathtt{\,-\,}}{\mathtt{85}}\right)}{{\mathtt{1\,000}}}}\\
\end{array} \right\} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = -{\mathtt{0.171\: \!058\: \!119\: \!895\: \!800\: \!7}}\\
{\mathtt{x}} = {\mathtt{0.001\: \!058\: \!119\: \!895\: \!800\: \!7}}\\
\end{array} \right\}$$

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Good answer NinjaDevo!

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It is a forward slash:  "/"  not a backward slash MG!

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Length of one side of the octagon must be 80/8 = 10 inches.

Angle of one side subtended at the centre = 360°/8 = 45°

Length of side of isosceles triangle formed by lines from centre to one side of the octagon = 5/sin(45/2°) inches.  This is also the radius of the circumscribed circle, so:

Area of circle = pi*(5/sin(22.5°))²  in²

$${\mathtt{Area}} = {\mathtt{\pi}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{5}}}{\underset{\,\,\,\,^{\textcolor[rgb]{0.66,0.66,0.66}{360^\circ}}}{{sin}}{\left({\mathtt{22.5}}^\circ\right)}}}\right)}^{{\mathtt{2}}} \Rightarrow {\mathtt{Area}} = {\mathtt{536.303\: \!412\: \!267\: \!149\: \!253\: \!5}}$$

Area ≈ 536.3 in²

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