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GeometryQuestion and Answers: Page 115

Question Number 5556 Answers: 0 Comments: 2

The area of a circle is diretly proportional to the square of its diameter. What is the constant of proportionality?

$$\mathrm{The}\:\mathrm{area}\:\mathrm{of}\:\mathrm{a}\:\mathrm{circle}\:\mathrm{is}\:\mathrm{diretly}\: \\ $$$$\mathrm{proportional}\:\mathrm{to}\:\mathrm{the}\:\mathrm{square}\:\mathrm{of} \\ $$$$\mathrm{its}\:\mathrm{diameter}. \\ $$$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{constant}\:\mathrm{of}\:\mathrm{proportionality}? \\ $$

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 5529 Answers: 0 Comments: 1

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 5490 Answers: 1 Comments: 0

∫(√(r^2 −x^2 ))dx=? ∫_(−r) ^x (√(r^2 −x^2 ))dx=?? ∫_x ^y (√(r^2 −x^2 ))dx=???

$$\int\sqrt{{r}^{\mathrm{2}} −{x}^{\mathrm{2}} }{dx}=? \\ $$$$\underset{−{r}} {\overset{{x}} {\int}}\sqrt{{r}^{\mathrm{2}} −{x}^{\mathrm{2}} }{dx}=?? \\ $$$$\underset{{x}} {\overset{{y}} {\int}}\sqrt{{r}^{\mathrm{2}} −{x}^{\mathrm{2}} }{dx}=??? \\ $$

Question Number 5467 Answers: 0 Comments: 0

⊠7n

$$\boxtimes\mathrm{7}{n} \\ $$

Question Number 5447 Answers: 1 Comments: 0

6/8

$$\mathrm{6}/\mathrm{8} \\ $$$$ \\ $$

Question Number 5441 Answers: 0 Comments: 4

Question Number 5417 Answers: 0 Comments: 2

Prove, or disprove, that the circle has the largest Perimiter over all natural shapes that have area A

$$\mathrm{Prove},\:\mathrm{or}\:\mathrm{disprove},\:\mathrm{that}\:\mathrm{the}\:\mathrm{circle} \\ $$$$\mathrm{has}\:\mathrm{the}\:\mathrm{largest}\:{Perimiter}\:\mathrm{over}\:\mathrm{all} \\ $$$$\mathrm{natural}\:\mathrm{shapes}\:\mathrm{that}\:\mathrm{have}\:\mathrm{area}\:{A} \\ $$

Question Number 5410 Answers: 0 Comments: 2

Question Number 5398 Answers: 0 Comments: 6

If we have an n−dimensional cube. How do we find its ′volume′?

$$\mathrm{If}\:\mathrm{we}\:\mathrm{have}\:\mathrm{an}\:{n}−\mathrm{dimensional}\:\mathrm{cube}. \\ $$$$\mathrm{How}\:\mathrm{do}\:\mathrm{we}\:\mathrm{find}\:\mathrm{its}\:'{volume}'? \\ $$

Question Number 5381 Answers: 0 Comments: 1

Question Number 5354 Answers: 1 Comments: 3

Question Number 5350 Answers: 0 Comments: 2

Suppose the radius of the earth is 6400km. the acceleration of a person at latitude 60° due to the earth rotation is ? (a) 0.034m/s^2 (b) 232.7m/s^2 (c) 0.0169m/s^2 (d) 465.4m/s^2 Please help. thanks in advance.

$${Suppose}\:{the}\:{radius}\:{of}\:{the}\:{earth}\:{is}\:\mathrm{6400}{km}.\:{the}\:{acceleration}\: \\ $$$${of}\:{a}\:{person}\:{at}\:{latitude}\:\mathrm{60}°\:{due}\:{to}\:{the}\:{earth}\:{rotation}\:{is}\:? \\ $$$$ \\ $$$$\left({a}\right)\:\mathrm{0}.\mathrm{034}{m}/{s}^{\mathrm{2}} \\ $$$$\left({b}\right)\:\mathrm{232}.\mathrm{7}{m}/{s}^{\mathrm{2}} \\ $$$$\left({c}\right)\:\mathrm{0}.\mathrm{0169}{m}/{s}^{\mathrm{2}} \\ $$$$\left({d}\right)\:\mathrm{465}.\mathrm{4}{m}/{s}^{\mathrm{2}} \\ $$$$ \\ $$$${Please}\:{help}.\:{thanks}\:{in}\:{advance}. \\ $$

Question Number 5341 Answers: 0 Comments: 2

Three circles with radius r The circles have equations: c_1 : x^2 +y^2 =r^2 c_2 : (x−r)^2 +y^2 =r^2 c_3 : x^2 +(y−r)^2 =r^2 Find the Areas of: 1. Enclosed area ABC 2. Enclosed area ACD

$$\mathrm{Three}\:\mathrm{circles}\:\mathrm{with}\:\mathrm{radius}\:{r} \\ $$$$\mathrm{The}\:\mathrm{circles}\:\mathrm{have}\:\mathrm{equations}: \\ $$$${c}_{\mathrm{1}} :\:\:\:\:\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} ={r}^{\mathrm{2}} \\ $$$${c}_{\mathrm{2}} :\:\:\:\:\:\left({x}−{r}\right)^{\mathrm{2}} +{y}^{\mathrm{2}} ={r}^{\mathrm{2}} \\ $$$${c}_{\mathrm{3}} :\:\:\:\:\:{x}^{\mathrm{2}} +\left({y}−{r}\right)^{\mathrm{2}} ={r}^{\mathrm{2}} \\ $$$$ \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{Areas}\:\mathrm{of}: \\ $$$$\mathrm{1}.\:\:\:\:\:\mathrm{Enclosed}\:\mathrm{area}\:{ABC} \\ $$$$\mathrm{2}.\:\:\:\:\:\mathrm{Enclosed}\:\mathrm{area}\:{ACD} \\ $$

Question Number 5338 Answers: 0 Comments: 1

Question Number 5334 Answers: 0 Comments: 1

A sphere of radius r contains a cyllinder of possible largest volume.Determine the volume of the cyllinder.

$$\mathrm{A}\:\boldsymbol{\mathrm{sphere}}\:\mathrm{of}\:\boldsymbol{\mathrm{radius}}\:\boldsymbol{\mathrm{r}}\:\:\mathrm{contains}\:\mathrm{a}\:\boldsymbol{\mathrm{cyllinder}}\:\mathrm{of}\:\mathrm{possible} \\ $$$$\mathrm{largest}\:\boldsymbol{\mathrm{volume}}.\boldsymbol{\mathrm{D}}\mathrm{etermine}\:\mathrm{the}\:\mathrm{volume}\:\mathrm{of}\:\mathrm{the}\:\boldsymbol{\mathrm{cyllinder}}. \\ $$

Question Number 5325 Answers: 0 Comments: 3

Question Number 5319 Answers: 1 Comments: 0

A sphere of radius r contains a cube inside it. All the vertices of the cube touch the surface of sphere. What is the volume of the cube?

$$\mathrm{A}\:\boldsymbol{\mathrm{sphere}}\:\mathrm{of}\:\mathrm{radius}\:\boldsymbol{\mathrm{r}}\:\mathrm{contains}\:\mathrm{a}\:\boldsymbol{\mathrm{cube}} \\ $$$$\mathrm{inside}\:\mathrm{it}.\:\mathrm{All}\:\mathrm{the}\:\boldsymbol{\mathrm{vertices}}\:\mathrm{of}\:\mathrm{the}\:\mathrm{cube} \\ $$$$\mathrm{touch}\:\mathrm{the}\:\boldsymbol{\mathrm{surface}}\:\mathrm{of}\:\mathrm{sphere}. \\ $$$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\boldsymbol{\mathrm{volume}}\:\mathrm{of}\:\mathrm{the}\:\mathrm{cube}? \\ $$

Question Number 5308 Answers: 0 Comments: 0

3 points with the restriction that they should be non-collinear determine circle. What number of points with what restriction determine sphere?

$$\mathrm{3}\:\mathrm{points}\:\mathrm{with}\:\mathrm{the}\:\mathrm{restriction}\:\mathrm{that} \\ $$$$\mathrm{they}\:\mathrm{should}\:\mathrm{be}\:\mathrm{non}-\mathrm{collinear}\:\mathrm{determine} \\ $$$$\mathrm{circle}. \\ $$$$\mathrm{What}\:\mathrm{number}\:\mathrm{of}\:\mathrm{points}\:\mathrm{with}\:\mathrm{what} \\ $$$$\mathrm{restriction}\:\mathrm{determine}\:\mathrm{sphere}? \\ $$

Question Number 5291 Answers: 0 Comments: 9

A circle has been drawn with one unit opened compass on surface of a sphere of one unit radius. What will be the area of the circle?

$$\mathrm{A}\:\mathrm{circle}\:\mathrm{has}\:\mathrm{been}\:\mathrm{drawn}\:\mathrm{with}\:\mathrm{one} \\ $$$$\mathrm{unit}\:\mathrm{opened}\:\mathrm{compass}\:\mathrm{on}\:\:\mathrm{surface} \\ $$$$\mathrm{of}\:\mathrm{a}\:\mathrm{sphere}\:\mathrm{of}\:\mathrm{one}\:\mathrm{unit}\:\mathrm{radius}.\: \\ $$$$\mathrm{What}\:\mathrm{will}\:\mathrm{be}\:\mathrm{the}\:\mathrm{area}\:\mathrm{of}\:\mathrm{the}\:\mathrm{circle}? \\ $$

Question Number 5290 Answers: 0 Comments: 0

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