All Answers 358411 Answers

Ask Francis. He'll know.

3+3*3-3+3

=3+9-3+3

=12

It's right if b= 0 !

What you have said is not correct!  Remember that raising to a power takes precedence over negation, so -3² without brackets is the same as -(3²).  Suppose this were not the case, then you could have the following strange situation:

-3² = 9

0 - 3² = -9

Yet most of us think of -3² as the same thing as 0 - 3².

If you don't like this convention (namely, that -3² is different from (-3)²), then use Microsoft Excel, which treats them as the same!  Most proper scientific calculators and mathematical software treat raising to the power as a higher priority than unary negation.

On the calculator on this site, if you enter -3², the calculator even puts brackets around the 3² to reinforce the fact that it is making the standard assumptions on hierarchy!

$${\mathtt{\,-\,}}\left({{\mathtt{3}}}^{{\mathtt{2}}}\right) = -{\mathtt{9}}$$

I thought that this would be a catenery - not a parabola.

I think that you have forgotten one of the answers heureka.

$$(2x+1)^2=16\\

2x+1=4\quad or \quad 2x+1=-4\\

2x=3\quad \:\:\:\:or \quad 2x=-5\\$$

$$x=1\frac{1}{2}\quad or \quad x=-2\frac{1}{2}\\$$

I was your looking at your LaTex code heureka,  I've not seen the \quad command before.

I used it in my answer.  Thank you.

Don't forget the other solution!

If (2x+1)² = 16 then 2x+1 = ±4

Taking the negative value we have 2x+1 = -4;  2x = -5;  x = -5/2 or x = -2.5

@@ End of Day Wrap - Tuesday 13/5/14   Sydney, Australia Time 19:40

Hi all,

Today was extremely busy, Chris (CPhil) and I were answering questions for many hours.  Many great answers were also forthcoming from Alan, BluePhoenix912, admin, zegroes, anonymous, cdromdj, GoldenLeaf, Creeperface and heureka.  Thank you everyone.  

I'd particularly like to thank Alan.  Every evening Alan comes to the forum and answers a number of questions, including the physics questions that are  beyond the other regular members' capabilities.  I really appreciate this. On behalf of all members I extend a very big thank you to Alan.

Kato99 also offered a nice show of appreciation.  Thanks for that.

Chris's first puzzle was solved by heureka and myself but Chris has changed the rules (Actually explained them better) so now the puzzle is really not solved.  There is a second easier puzzle here as well.

$$\\(2x+1)^2=16\quad|\quad\sqrt{}\\
2x+1=4\quad|\quad-1\\
2x=3\quad|\quad:2\\
x=\frac{3}{2} \quad or \quad x=1.5$$

Food = 0.3X

left = 0.7X

stationary = 0.05*0.7X = 0.035X

Food Plus stationary = 0.335X

Left for banking = X-0.335X = (1-0.335)X = 0.665X

how to make a regular variable for example, how would I make x+1 sq root x to the 4th power - 1

$$x+1 (\sqrt x)^4-1$$

Okay, I do not understand your question? Is this what your words mean?

Maybe because it explains things that we can't actually see.

Thanks Alan.

To find the sample mean add them all up and divide by how many there are:

$${\mathtt{mean}} = {\frac{\left({\mathtt{15}}{\mathtt{\,\small\textbf+\,}}{\mathtt{18}}{\mathtt{\,\small\textbf+\,}}{\mathtt{16}}{\mathtt{\,\small\textbf+\,}}{\mathtt{19}}{\mathtt{\,\small\textbf+\,}}{\mathtt{14}}{\mathtt{\,\small\textbf+\,}}{\mathtt{12}}{\mathtt{\,\small\textbf+\,}}{\mathtt{14}}{\mathtt{\,\small\textbf+\,}}{\mathtt{12}}{\mathtt{\,\small\textbf+\,}}{\mathtt{17}}{\mathtt{\,\small\textbf+\,}}{\mathtt{15}}{\mathtt{\,\small\textbf+\,}}{\mathtt{11}}\right)}{{\mathtt{11}}}} \Rightarrow {\mathtt{mean}} = {\mathtt{14.818\: \!181\: \!818\: \!181\: \!818\: \!2}}$$

To find the sample standard deviation use the following formula:

$$sd = \sqrt{\frac{\sum_{i=1}^n(x_i-mean)^2}{n-1}}$$

where n is the number of items (11 here), xi are the individual data values (15, 18 ... etc.), and $$\sum_{i=1}^n$$ means sum all of the squared bracketed terms that follow.  

I'll leave you to plug in the numbers.

You give no information about what comparison is to be done, so there is no way I can say anything about what the null hypothesis and its alternate might be! 

Anonymous's answer assumed the radius is the apothem, which is the radius of the maximum inscribed circle.

If the radius is that of the minimum circumscribed circle (imagine a circle completely surrounding the polygon and shrinking it until it just touches all the vertices, but doesn't cross to the inside of the polygon anywhere) then, assuming we have a regular polygon with n sides, chop the polygon into n equal triangles.  Find the area of each triangle from (1/2)base*height, where base = 2*radius*sin(180°/n) and height = radius*cos(180°/n).

So total area of polygon = n*radius²*sin(180°/n)*cos(180°/n) = (n/2)*radius²*sin(360°/n)

Thanks Alan.

I don't remember ever seeing this identity before!  

Thanks Chris.   

Frictional force is given by $$F=\mu W$$ where μ is the coefficient of friction and W is the weight (13552.2N here).

The centripetal force required to prevent slipping is $$F=m\frac{v^2}{r}$$ where m is the mass (=W/g), v is speed and r is radius.

Equating these two forces we have: $$\mu W = \frac{W}{g}\frac{v^2}{r}$$  so  $$\mu = \frac{v^2}{gr}$$  

v = 52.2*10³/3600 m/s, r = 1.68*102m and g = 9.81m/s². 

$${\frac{{\left({\frac{{\mathtt{52\,200}}}{{\mathtt{3\,600}}}}\right)}^{{\mathtt{2}}}}{\left({\mathtt{9.81}}{\mathtt{\,\times\,}}{\mathtt{1.68}}{\mathtt{\,\times\,}}{\mathtt{102}}\right)}} = {\mathtt{0.125\: \!071\: \!265\: \!339\: \!299\: \!2}}$$

So  $$\mu = 0.125$$  approx.

What are you doing GoldenLeaf?

Some of what you did made some sense.

First I'll simplify it by dividing everything by 25

$$25y^2 - 1600x^2 = 40000 \\

y^2-64x^2=1600$$

Now if x=0 then yes y=+or-40

If y=0 there are no solutions for x which means that the graph will not cross the x axis.

There is not going to be only 1 solution because there are 2 variables and only 1 equation.

you would need two eqations in order to have only one solution.

So yes there will be an infinite number of solutions .

Every point on this graph will be a solution.

I hope that helps!

That is not correct GoldenLeaf

$$(x + 3.2)^2 + (y + 3.5)^2\\\\
=x^2+(2*3.2x)+3.2^2 + y^2+(2*3.5y)+3.5^2\\\\
=x^2+6.4x+10.24 + y^2+7y+12.25\\\\
=x^2+6.4x+y^2+7y+22.49\\\\$$

Here is a video.

I won't rave about it - it is not the way I would usually explain it - but you might find it good.

$$\\\text{a.)}\;0.300*x=30\%*x\\
\text{b.)}\;0.035*x=(1-0.3)*0.05*x=(1-30\%)*5\%*x\\
\text{c.)}\;0.665*x=(1-0.05)*0.7*x=(1-5\%)*(1-30\%)*x\\
\sum =1.000*x$$

I think CPhill has assumed that the base is an equilateral triangle of side length 18 units.

This is not actually stated in the question but i guess it is a reasonable assumption.

All the angels are 60 degrees

----------------------

in triangle ABC 

 $${\mathtt{Area}} = {\frac{{\mathtt{1}}}{{\mathtt{2}}}}{\mathtt{\,\times\,}}{\mathtt{a}}{\mathtt{\,\times\,}}{\mathtt{b}}{\mathtt{\,\times\,}}{\mathtt{sinC}}$$

$${\mathtt{Area}} = {\frac{{\mathtt{1}}}{{\mathtt{2}}}}{\mathtt{\,\times\,}}{\mathtt{18}}{\mathtt{\,\times\,}}{\mathtt{18}}{\mathtt{\,\times\,}}{\mathtt{sin60}}$$

$${\mathtt{Area}} = {\frac{{\mathtt{162}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{3}}}}}{{\mathtt{2}}}}$$

$${\mathtt{Area}} = {\mathtt{81}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{3}}}}$$

---------------------------------------

$${\mathtt{Volume}} = {\frac{{\mathtt{1}}}{{\mathtt{3}}}}{\mathtt{\,\times\,}}{\mathtt{Area}}{\mathtt{\,\times\,}}{\mathtt{height}}$$

$${\mathtt{Volume}} = {\frac{{\mathtt{1}}}{{\mathtt{3}}}}{\mathtt{\,\times\,}}{\mathtt{81}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{3}}}}{\mathtt{\,\times\,}}{\mathtt{4}}$$

$${\mathtt{Volume}} = {\mathtt{108}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{3}}}}$$    units cubed.   

$$Volume \approx 187unit^3$$

Only the x is raised to the negative power not the 2

remember PEDMAS

There is an invisable x sign between the 2 and the x 

and, you must do exponents (powers) before you do multiply.  so only the x is raised to the power of -2