Question-146457

Question Number 146457 by mathdanisur last updated on 13/Jul/21

Commented by mathdanisur last updated on 13/Jul/21

$${find}\:\:{x}=? \\ $$

Answered by gsk2684 last updated on 13/Jul/21

let ∠BAC =θ using cosine rule 7^2 =x^2 +2^2 −2.x.2.cos (90^0 +θ) 49=x^2 +4+4x sin θ 45=x^2 +4x((x/(23))) 45×23=23x^2 +4x^2 x^2 =((45×23)/(27))=((115)/3) x=(√((115)/3))

$${let}\:\angle{BAC}\:=\theta \\ $$$${using}\:{cosine}\:{rule}\: \\ $$$$\mathrm{7}^{\mathrm{2}} ={x}^{\mathrm{2}} +\mathrm{2}^{\mathrm{2}} −\mathrm{2}.{x}.\mathrm{2}.\mathrm{cos}\:\left(\mathrm{90}^{\mathrm{0}} +\theta\right) \\ $$$$\mathrm{49}={x}^{\mathrm{2}} +\mathrm{4}+\mathrm{4}{x}\:\mathrm{sin}\:\theta \\ $$$$\mathrm{45}={x}^{\mathrm{2}} +\mathrm{4}{x}\left(\frac{{x}}{\mathrm{23}}\right) \\ $$$$\mathrm{45}×\mathrm{23}=\mathrm{23}{x}^{\mathrm{2}} +\mathrm{4}{x}^{\mathrm{2}} \\ $$$${x}^{\mathrm{2}} =\frac{\mathrm{45}×\mathrm{23}}{\mathrm{27}}=\frac{\mathrm{115}}{\mathrm{3}} \\ $$$${x}=\sqrt{\frac{\mathrm{115}}{\mathrm{3}}} \\ $$

Commented by mathdanisur last updated on 13/Jul/21

$${thank}\:{you}\:{Ser}\:{cool} \\ $$

Answered by mr W last updated on 15/Jul/21

x^2 =2^2 +7^2 −2×2×7 cos D ...(i) 23^2 −x^2 =25^2 +7^2 −2×25×7 cos D ...(ii) (i)×25−(ii)×2: 25x^2 −2×23^2 +2x^2 =25(2^2 +7^2 )−2(25^2 +7^2 ) 27x^2 =45×23 x=((√(345))/3)

$${x}^{\mathrm{2}} =\mathrm{2}^{\mathrm{2}} +\mathrm{7}^{\mathrm{2}} −\mathrm{2}×\mathrm{2}×\mathrm{7}\:\mathrm{cos}\:{D}\:\:\:\:…\left({i}\right) \\ $$$$\mathrm{23}^{\mathrm{2}} −{x}^{\mathrm{2}} =\mathrm{25}^{\mathrm{2}} +\mathrm{7}^{\mathrm{2}} −\mathrm{2}×\mathrm{25}×\mathrm{7}\:\mathrm{cos}\:{D}\:\:\:…\left({ii}\right) \\ $$$$\left({i}\right)×\mathrm{25}−\left({ii}\right)×\mathrm{2}: \\ $$$$\mathrm{25}{x}^{\mathrm{2}} −\mathrm{2}×\mathrm{23}^{\mathrm{2}} +\mathrm{2}{x}^{\mathrm{2}} =\mathrm{25}\left(\mathrm{2}^{\mathrm{2}} +\mathrm{7}^{\mathrm{2}} \right)−\mathrm{2}\left(\mathrm{25}^{\mathrm{2}} +\mathrm{7}^{\mathrm{2}} \right) \\ $$$$\mathrm{27}{x}^{\mathrm{2}} =\mathrm{45}×\mathrm{23} \\ $$$${x}=\frac{\sqrt{\mathrm{345}}}{\mathrm{3}} \\ $$

Commented by mathdanisur last updated on 19/Jul/21

$${thank}\:{you}\:{Ser},\:{but}\:{answer}:\:\:\frac{\sqrt{\mathrm{115}}}{\mathrm{3}} \\ $$

Commented by mr W last updated on 20/Jul/21

you should check before you say! where is wrong in my working? my answer is ((√(345))/3). if you mean the answer is (√((115)/3)), then ((√(345))/3) is also right.

$${you}\:{should}\:{check}\:{before}\:{you}\:{say}! \\ $$$${where}\:{is}\:{wrong}\:{in}\:{my}\:{working}? \\ $$$${my}\:{answer}\:{is}\:\frac{\sqrt{\mathrm{345}}}{\mathrm{3}}.\:{if}\:{you}\:{mean}\:{the} \\ $$$${answer}\:{is}\:\sqrt{\frac{\mathrm{115}}{\mathrm{3}}},\:{then}\:\frac{\sqrt{\mathrm{345}}}{\mathrm{3}}\:{is}\:{also} \\ $$$${right}. \\ $$

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