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Question Number 174591 Answers: 1 Comments: 0

The drawing below shows two equilateral triangles with side length a. The triangle are horizontally shifted by (a/2). Find the intersection area A of the two triangles (grey area).

$$\mathrm{The}\:\mathrm{drawing}\:\mathrm{below}\:\mathrm{shows}\:\mathrm{two}\: \\ $$$$\mathrm{equilateral}\:\mathrm{triangles}\:\mathrm{with}\:\mathrm{side}\:\mathrm{length} \\ $$$$\boldsymbol{\mathrm{a}}.\:\mathrm{The}\:\mathrm{triangle}\:\mathrm{are}\:\mathrm{horizontally}\:\mathrm{shifted} \\ $$$$\mathrm{by}\:\frac{\mathrm{a}}{\mathrm{2}}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{intersection}\:\mathrm{area}\:\mathrm{A}\:\mathrm{of} \\ $$$$\mathrm{the}\:\mathrm{two}\:\mathrm{triangles}\:\left(\mathrm{grey}\:\mathrm{area}\right). \\ $$$$ \\ $$$$ \\ $$$$ \\ $$

Question Number 174587 Answers: 0 Comments: 2

Question Number 174583 Answers: 0 Comments: 0

Your friend makes 6 field goals and 2 free throws. You make twice as many field goals as your friend and half the number of free throws. How many points do you have. explain the order of operation you followed

$$\:\mathrm{Your}\:\mathrm{friend}\:\mathrm{makes}\:\mathrm{6}\:\mathrm{field}\:\mathrm{goals}\:\mathrm{and}\: \\ $$$$\:\:\mathrm{2}\:\mathrm{free}\:\mathrm{throws}.\:\mathrm{You}\:\mathrm{make}\:\mathrm{twice}\:\mathrm{as} \\ $$$$\:\mathrm{many}\:\mathrm{field}\:\mathrm{goals}\:\mathrm{as}\:\mathrm{your}\:\mathrm{friend}\:\mathrm{and}\: \\ $$$$\:\mathrm{half}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{free}\:\mathrm{throws}.\:\mathrm{How} \\ $$$$\mathrm{many}\:\mathrm{points}\:\mathrm{do}\:\mathrm{you}\:\mathrm{have}.\:\mathrm{explain}\:\mathrm{the} \\ $$$$\mathrm{order}\:\mathrm{of}\:\mathrm{operation}\:\mathrm{you}\:\mathrm{followed} \\ $$$$\:\:\:\:\: \\ $$

Question Number 174582 Answers: 0 Comments: 0

Your friend makes 4 field goals . you make three times as many field goals as your friend plus one field goals. how many points do you have. Explain the order of operation you followed

$$\:\mathrm{Your}\:\mathrm{friend}\:\mathrm{makes}\:\mathrm{4}\:\mathrm{field}\:\mathrm{goals}\:.\:\mathrm{you} \\ $$$$\:\mathrm{make}\:\mathrm{three}\:\mathrm{times}\:\mathrm{as}\:\mathrm{many}\:\mathrm{field}\:\mathrm{goals} \\ $$$$\mathrm{as}\:\mathrm{your}\:\mathrm{friend}\:\mathrm{plus}\:\mathrm{one}\:\mathrm{field}\:\mathrm{goals}. \\ $$$$\:\mathrm{how}\:\mathrm{many}\:\mathrm{points}\:\mathrm{do}\:\mathrm{you}\:\mathrm{have}.\: \\ $$$$\mathrm{Explain}\:\mathrm{the}\:\mathrm{order}\:\mathrm{of}\:\mathrm{operation}\:\mathrm{you}\: \\ $$$$\mathrm{followed} \\ $$

Question Number 174581 Answers: 1 Comments: 0

3^3^3^3 find last two digits

$$\mathrm{3}^{\mathrm{3}^{\mathrm{3}^{\mathrm{3}} } } \\ $$$$\boldsymbol{\mathrm{find}}\:\boldsymbol{\mathrm{last}}\:\boldsymbol{\mathrm{two}}\:\boldsymbol{\mathrm{digits}} \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$

Question Number 174579 Answers: 0 Comments: 0

find minimum value of: ((a^2 (b−1)^2 +2ab+(a−1)^2 )/(a(ab+1))) [all∈R]

$$\:\:\:\:{find}\:{minimum}\:{value}\:{of}: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\frac{\boldsymbol{{a}}^{\mathrm{2}} \left(\boldsymbol{{b}}−\mathrm{1}\right)^{\mathrm{2}} +\mathrm{2}\boldsymbol{{ab}}+\left(\boldsymbol{{a}}−\mathrm{1}\right)^{\mathrm{2}} }{\boldsymbol{{a}}\left(\boldsymbol{{ab}}+\mathrm{1}\right)}\:\:\:\:\:\:\:\:\:\left[\boldsymbol{{all}}\in\boldsymbol{{R}}\right] \\ $$

Question Number 174578 Answers: 0 Comments: 0

Question Number 174576 Answers: 1 Comments: 0

Question Number 174570 Answers: 1 Comments: 1

{ ((f((x/(x^2 +1)))=(x^2 /(x^4 +2x^2 +1)))),((f((√3) )=?)) :}

$$\:\:\:\:\:\:\begin{cases}{\boldsymbol{{f}}\left(\frac{\boldsymbol{{x}}}{\boldsymbol{{x}}^{\mathrm{2}} +\mathrm{1}}\right)=\frac{\boldsymbol{{x}}^{\mathrm{2}} }{\boldsymbol{{x}}^{\mathrm{4}} +\mathrm{2}\boldsymbol{{x}}^{\mathrm{2}} +\mathrm{1}}}\\{\boldsymbol{{f}}\left(\sqrt{\mathrm{3}}\:\right)=?}\end{cases} \\ $$

Question Number 174569 Answers: 0 Comments: 0

{ ((x+z=0)),((y+t+xz=−2b)),((xt+yz=1)),((yt=b^2 −a)) :} [ all∈R] solve for : x, y ,z, t .

$$\:\:\:\:\:\begin{cases}{\boldsymbol{{x}}+\boldsymbol{{z}}=\mathrm{0}}\\{\boldsymbol{{y}}+\boldsymbol{{t}}+\boldsymbol{{xz}}=−\mathrm{2}\boldsymbol{{b}}}\\{\boldsymbol{{xt}}+\boldsymbol{{yz}}=\mathrm{1}}\\{\boldsymbol{{yt}}=\boldsymbol{{b}}^{\mathrm{2}} −\boldsymbol{{a}}}\end{cases}\:\:\:\:\:\:\left[\:\:\boldsymbol{{all}}\in\boldsymbol{{R}}\right] \\ $$$$\boldsymbol{{solve}}\:\boldsymbol{{for}}\:\:\::\:\:\:\boldsymbol{{x}},\:\boldsymbol{{y}}\:,\boldsymbol{{z}},\:\boldsymbol{{t}}\:\:\:\:\:\:\:\:\:\:. \\ $$

Question Number 174568 Answers: 1 Comments: 0

If (1/a) + (1/b) + (1/c) = (1/(a + b + c)) then (1/a^3 ) + (1/(b^3 )) + (1/c^3 ) = ?

$$\mathrm{If}\:\frac{\mathrm{1}}{{a}}\:+\:\frac{\mathrm{1}}{{b}}\:+\:\frac{\mathrm{1}}{{c}}\:=\:\frac{\mathrm{1}}{{a}\:+\:{b}\:+\:{c}}\:\mathrm{then}\: \\ $$$$\frac{\mathrm{1}}{{a}^{\mathrm{3}} }\:+\:\frac{\mathrm{1}}{{b}^{\mathrm{3}} \:}\:+\:\frac{\mathrm{1}}{{c}^{\mathrm{3}} }\:=\:? \\ $$

Question Number 174567 Answers: 0 Comments: 0

Question Number 174566 Answers: 1 Comments: 0

Question Number 174560 Answers: 2 Comments: 0

Ω=∫_0 ^( 1) (( dx)/( (√(8+3x−(√(1+( x^( 2) +3x +2)(x^( 2) +7x+12)))))))

$$ \\ $$$$\:\:\:\Omega=\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\:{dx}}{\:\sqrt{\mathrm{8}+\mathrm{3}{x}−\sqrt{\mathrm{1}+\left(\:{x}^{\:\mathrm{2}} +\mathrm{3}{x}\:+\mathrm{2}\right)\left({x}^{\:\mathrm{2}} +\mathrm{7}{x}+\mathrm{12}\right)}}} \\ $$$$ \\ $$

Question Number 174556 Answers: 2 Comments: 1

Question Number 174552 Answers: 2 Comments: 3

Question Number 174548 Answers: 1 Comments: 0

Evaluate . 𝛀 = ∫_0 ^( 2) (( e^( x) )/(e^( 1−x) + e^( x−1) )) dx= ?

$$ \\ $$$$\boldsymbol{{Evaluate}}\:. \\ $$$$\:\:\boldsymbol{\Omega}\:=\:\int_{\mathrm{0}} ^{\:\mathrm{2}} \frac{\:\:\boldsymbol{{e}}^{\:\boldsymbol{{x}}} }{\boldsymbol{{e}}^{\:\mathrm{1}−\boldsymbol{{x}}} +\:\boldsymbol{{e}}^{\:\boldsymbol{{x}}−\mathrm{1}} }\:\boldsymbol{{dx}}=\:? \\ $$$$\:\:\:\: \\ $$

Question Number 174538 Answers: 1 Comments: 3

what is the maximum and minimum value of “x” if x+y+z = 6, x^2 +y^2 +z^2 = 18, for real values of “x,y and z”

$$\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{maximum}\:\mathrm{and} \\ $$$$\mathrm{minimum}\:\mathrm{value}\:\mathrm{of}\:``\mathrm{x}''\:\mathrm{if} \\ $$$$\mathrm{x}+\mathrm{y}+\mathrm{z}\:=\:\mathrm{6},\:\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} +\mathrm{z}^{\mathrm{2}} \:=\:\mathrm{18}, \\ $$$$\mathrm{for}\:\mathrm{real}\:\mathrm{values}\:\mathrm{of}\:``\mathrm{x},\mathrm{y}\:\mathrm{and}\:\mathrm{z}'' \\ $$

Question Number 174534 Answers: 0 Comments: 0

Question Number 174533 Answers: 1 Comments: 0

Question Number 174532 Answers: 0 Comments: 0

Question Number 174528 Answers: 1 Comments: 0

Question Number 174519 Answers: 1 Comments: 2

Question Number 174522 Answers: 0 Comments: 0

Let σ(n) be the sum of all positive divisors of the integer n and let p be any prime number. Show that σ(n)<2n holds true for all n of the form n=p^2 . Mastermind

$$\mathrm{Let}\:\sigma\left(\mathrm{n}\right)\:\mathrm{be}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{all}\:\mathrm{positive}\:\mathrm{divisors} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{integer}\:\mathrm{n}\:\mathrm{and}\:\mathrm{let}\:\mathrm{p}\:\mathrm{be}\:\mathrm{any}\:\mathrm{prime} \\ $$$$\mathrm{number}.\:\mathrm{Show}\:\mathrm{that}\:\sigma\left(\mathrm{n}\right)<\mathrm{2n}\:\mathrm{holds}\:\mathrm{true} \\ $$$$\mathrm{for}\:\mathrm{all}\:\mathrm{n}\:\mathrm{of}\:\mathrm{the}\:\mathrm{form}\:\mathrm{n}=\mathrm{p}^{\mathrm{2}} . \\ $$$$ \\ $$$$\mathrm{Mastermind} \\ $$

Question Number 174514 Answers: 3 Comments: 1

a+b+c=1 a^2 +b^2 +c^2 =2 a^3 +b^3 +c^3 =3 then a^5 +b^5 +c^5 ?

$${a}+{b}+{c}=\mathrm{1} \\ $$$${a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} =\mathrm{2} \\ $$$${a}^{\mathrm{3}} +{b}^{\mathrm{3}} +{c}^{\mathrm{3}} =\mathrm{3} \\ $$$${then}\:{a}^{\mathrm{5}} +{b}^{\mathrm{5}} +{c}^{\mathrm{5}} \:? \\ $$

Question Number 174512 Answers: 1 Comments: 0

how many integer a,b∈z^+ a^5 −b^5 =10(b+1)^2 −9

$${how}\:{many}\:{integer}\:{a},{b}\in{z}^{+} \\ $$$${a}^{\mathrm{5}} −{b}^{\mathrm{5}} =\mathrm{10}\left({b}+\mathrm{1}\right)^{\mathrm{2}} −\mathrm{9} \\ $$$$ \\ $$

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