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Question Number 5546 Answers: 0 Comments: 13

I have two circles side by side. Circle 1 has radius r. Circle 2 has radius (1/3)r. If I roll circle 2 around circle 1 until it reaches the begining, how many times will it roll?

$$\mathrm{I}\:\mathrm{have}\:\mathrm{two}\:\mathrm{circles}\:\mathrm{side}\:\mathrm{by}\:\mathrm{side}. \\ $$$$\mathrm{Circle}\:\mathrm{1}\:\mathrm{has}\:\mathrm{radius}\:{r}. \\ $$$$\mathrm{Circle}\:\mathrm{2}\:\mathrm{has}\:\mathrm{radius}\:\frac{\mathrm{1}}{\mathrm{3}}{r}. \\ $$$$ \\ $$$$\mathrm{If}\:\mathrm{I}\:\mathrm{roll}\:\mathrm{circle}\:\mathrm{2}\:\mathrm{around}\:\mathrm{circle}\:\mathrm{1}\:\mathrm{until} \\ $$$$\mathrm{it}\:\mathrm{reaches}\:\mathrm{the}\:\mathrm{begining}, \\ $$$$\mathrm{how}\:\mathrm{many}\:\mathrm{times}\:\mathrm{will}\:\mathrm{it}\:\mathrm{roll}? \\ $$

Question Number 5543 Answers: 0 Comments: 0

•What plane geometrical figures could be produced by joining vertices of a cube? (for example : square) •What largest area(2 dimensional) can be obtained by joining vertices of a cube when its side is x?

$$\bullet\mathrm{What}\:\boldsymbol{\mathrm{plane}}\:\boldsymbol{\mathrm{geometrical}}\:\boldsymbol{\mathrm{figures}} \\ $$$$\mathrm{could}\:\mathrm{be}\:\mathrm{produced}\:\mathrm{by}\:\mathrm{joining} \\ $$$$\boldsymbol{\mathrm{vertices}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{a}}\:\boldsymbol{\mathrm{cube}}? \\ $$$$\left(\mathrm{for}\:\mathrm{example}\::\:\mathrm{square}\right) \\ $$$$\bullet\mathrm{What}\:\mathrm{largest}\:\mathrm{area}\left(\mathrm{2}\:\mathrm{dimensional}\right)\:\mathrm{can}\:\mathrm{be}\:\mathrm{obtained} \\ $$$$\mathrm{by}\:\mathrm{joining}\:\mathrm{vertices}\:\mathrm{of}\:\mathrm{a}\:\mathrm{cube}\:\mathrm{when}\:\mathrm{its}\:\mathrm{side}\:\mathrm{is}\:\mathrm{x}? \\ $$

Question Number 5542 Answers: 0 Comments: 1

The surface area of a cylinder is: A=2πr^2 +2πrh As h→0, the shape becomes a 2D circle, so should lim_(h→0) A = πr^2 ??? Or is it that as h→0, it creates a circle of 3 dimensions with infintesimally small height, so it has two circles making it: lim_(h→0) A = 2πr^2 ???

$$\mathrm{The}\:\mathrm{surface}\:\mathrm{area}\:\mathrm{of}\:\mathrm{a}\:\mathrm{cylinder}\:\mathrm{is}: \\ $$$${A}=\mathrm{2}\pi{r}^{\mathrm{2}} +\mathrm{2}\pi{rh} \\ $$$$ \\ $$$$\mathrm{As}\:{h}\rightarrow\mathrm{0},\:\mathrm{the}\:\mathrm{shape}\:\mathrm{becomes}\:\mathrm{a}\:\mathrm{2D}\:\mathrm{circle}, \\ $$$$\mathrm{so}\:\mathrm{should}\:\:\:\:\:\underset{{h}\rightarrow\mathrm{0}} {\mathrm{lim}}\:{A}\:=\:\pi{r}^{\mathrm{2}} \:\:\:??? \\ $$$$\mathrm{Or}\:\mathrm{is}\:\mathrm{it}\:\mathrm{that}\:\mathrm{as}\:{h}\rightarrow\mathrm{0},\:\mathrm{it}\:\mathrm{creates}\:\mathrm{a}\:\mathrm{circle} \\ $$$$\mathrm{of}\:\mathrm{3}\:\mathrm{dimensions}\:\mathrm{with}\:\mathrm{infintesimally}\:\mathrm{small} \\ $$$$\mathrm{height},\:\mathrm{so}\:\mathrm{it}\:\mathrm{has}\:\mathrm{two}\:\mathrm{circles}\:\mathrm{making}\:\mathrm{it}: \\ $$$$\underset{{h}\rightarrow\mathrm{0}} {\mathrm{lim}}\:{A}\:=\:\mathrm{2}\pi{r}^{\mathrm{2}} \:\:\:\:??? \\ $$

Question Number 5539 Answers: 0 Comments: 0

How many planes can pass through vertices of a cube, a) when each plane contains at least three vertices. b) when each plane contains exactly four vertices. c) Could we say that a plane can pass at most four vertices?

Question Number 5533 Answers: 0 Comments: 1

Measure of the side of a cube is double of the the side of the cube inside it. What is empty space in larger cube?

$$\mathrm{Measure}\:\mathrm{of}\:\mathrm{the}\:\mathrm{side}\:\mathrm{of}\:\mathrm{a}\:\mathrm{cube}\:\mathrm{is}\:\mathrm{double} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{the}\:\mathrm{side}\:\mathrm{of}\:\mathrm{the}\:\mathrm{cube}\:\mathrm{inside}\:\mathrm{it}. \\ $$$$\mathrm{What}\:\mathrm{is}\:\mathrm{empty}\:\mathrm{space}\:\mathrm{in}\:\mathrm{larger}\:\mathrm{cube}? \\ $$

Question Number 5532 Answers: 0 Comments: 1

Volume of a cube and its surface-area are numerically equal.What is the measure of the side of the cube?

$$\mathrm{Volume}\:\mathrm{of}\:\mathrm{a}\:\mathrm{cube}\:\mathrm{and}\:\mathrm{its}\:\mathrm{surface}-\mathrm{area} \\ $$$$\mathrm{are}\:\mathrm{numerically}\:\mathrm{equal}.\mathrm{What}\:\mathrm{is}\:\mathrm{the} \\ $$$$\mathrm{measure}\:\mathrm{of}\:\mathrm{the}\:\mathrm{side}\:\mathrm{of}\:\mathrm{the}\:\mathrm{cube}? \\ $$$$ \\ $$

Question Number 5531 Answers: 0 Comments: 2

Once I opened an algebra-book.Instead of individual page numbers there was written:“sum of page numbers infront of you is x”.Determine the page numbers in terms of x.

$$\mathrm{Once}\:\mathrm{I}\:\mathrm{opened}\:\mathrm{an}\:\mathrm{algebra}-\mathrm{book}.\mathrm{Instead} \\ $$$$\mathrm{of}\:\mathrm{individual}\:\mathrm{page}\:\mathrm{numbers}\:\mathrm{there}\:\mathrm{was} \\ $$$$\mathrm{written}:``{sum}\:{of}\:{page}\:{numbers}\:\:{infront} \\ $$$${of}\:{you}\:{is}\:\mathrm{x}''.\mathrm{Determine}\:\mathrm{the}\:\mathrm{page}\:\mathrm{numbers} \\ $$$$\mathrm{in}\:\mathrm{terms}\:\mathrm{of}\:\mathrm{x}. \\ $$

Question Number 5529 Answers: 0 Comments: 1

Question Number 5525 Answers: 0 Comments: 1

Expand f(z) = cos(z) about point z = Π/3

$${Expand}\:\:{f}\left({z}\right)\:=\:{cos}\left({z}\right) \\ $$$${about}\:{point}\:\:{z}\:=\:\Pi/\mathrm{3} \\ $$$$ \\ $$$$ \\ $$

Question Number 5522 Answers: 1 Comments: 0

3^x +3^(3−x) −18=10 x=?

$$\mathrm{3}^{{x}} +\mathrm{3}^{\mathrm{3}−{x}} −\mathrm{18}=\mathrm{10} \\ $$$${x}=? \\ $$

Question Number 5519 Answers: 0 Comments: 3

P=Π_(i=1) ^n (a^i /b^i ) a≠b, b≠0 P=??

$${P}=\underset{{i}=\mathrm{1}} {\overset{{n}} {\prod}}\frac{{a}^{{i}} }{{b}^{{i}} }\:\:\:\:\:\:\:\:{a}\neq{b},\:{b}\neq\mathrm{0} \\ $$$${P}=?? \\ $$

Question Number 5517 Answers: 0 Comments: 1

Question Number 5515 Answers: 1 Comments: 1

If you have a regular n−sided polygon, is there a method to calculate the area from one corner to another? That is, if we start at a corner (corner 1), and draw a line to corner x, what is the area? See image in comment for visual representation.

$$\mathrm{If}\:\mathrm{you}\:\mathrm{have}\:\mathrm{a}\:\mathrm{regular}\:{n}−\mathrm{sided}\:\mathrm{polygon}, \\ $$$$\mathrm{is}\:\mathrm{there}\:\mathrm{a}\:\mathrm{method}\:\mathrm{to}\:\mathrm{calculate}\:\mathrm{the}\:\mathrm{area} \\ $$$$\mathrm{from}\:\mathrm{one}\:\mathrm{corner}\:\mathrm{to}\:\mathrm{another}? \\ $$$$ \\ $$$$\mathrm{That}\:\mathrm{is},\:\mathrm{if}\:\mathrm{we}\:\mathrm{start}\:\mathrm{at}\:\mathrm{a}\:\mathrm{corner}\:\left(\mathrm{corner}\:\mathrm{1}\right), \\ $$$$\mathrm{and}\:\mathrm{draw}\:\mathrm{a}\:\mathrm{line}\:\mathrm{to}\:\mathrm{corner}\:{x},\:\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{area}? \\ $$$$\mathrm{See}\:\mathrm{image}\:\mathrm{in}\:\mathrm{comment}\:\mathrm{for}\:\mathrm{visual}\:\mathrm{representation}. \\ $$

Question Number 5513 Answers: 0 Comments: 0

If n>1, prove by mathematical induction that n((n+1)^(1/n) −1) < 1+(1/2)+(1/3)+(1/4)+...(1/n).

$${If}\:{n}>\mathrm{1},\:{prove}\:{by}\:{mathematical}\:{induction}\:{that} \\ $$$${n}\left(\left({n}+\mathrm{1}\right)^{\frac{\mathrm{1}}{{n}}} −\mathrm{1}\right)\:<\:\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{3}}+\frac{\mathrm{1}}{\mathrm{4}}+...\frac{\mathrm{1}}{{n}}. \\ $$

Question Number 5591 Answers: 0 Comments: 1

For x^x^x^(...) =a ∴ x^a =a ⇒ x=a^(1/a) Is this true? When is it not true?

$$\mathrm{For}\:{x}^{{x}^{{x}^{...} } } ={a} \\ $$$$\therefore\:{x}^{{a}} ={a}\:\:\:\Rightarrow\:\:\:{x}={a}^{\frac{\mathrm{1}}{{a}}} \\ $$$$ \\ $$$$\mathrm{Is}\:\mathrm{this}\:\mathrm{true}? \\ $$$$\mathrm{When}\:\mathrm{is}\:\mathrm{it}\:\mathrm{not}\:\mathrm{true}? \\ $$

Question Number 5508 Answers: 1 Comments: 0

24÷56

$$\mathrm{24}\boldsymbol{\div}\mathrm{56} \\ $$

Question Number 5507 Answers: 1 Comments: 0

if (x−1) (x+1) +3x+x^2 =ax^2 +bx+c then a=? b=? c=?