Question-209639

Question Number 209639 by SonGoku last updated on 17/Jul/24

Commented by SonGoku last updated on 17/Jul/24

In the figure, DC//FG is the width of a river. How wideis this river? (How do I solve it?)

$$\mathrm{In}\:\mathrm{the}\:\mathrm{figure},\:\mathrm{DC}//\mathrm{FG}\:\mathrm{is}\:\mathrm{the}\:\mathrm{width}\:\mathrm{of}\:\mathrm{a}\:\mathrm{river}. \\ $$$$\mathrm{How}\:\mathrm{wideis}\:\mathrm{this}\:\mathrm{river}? \\ $$$$\left(\mathrm{How}\:\mathrm{do}\:\mathrm{I}\:\mathrm{solve}\:\mathrm{it}?\right) \\ $$

Answered by mr W last updated on 17/Jul/24

Commented by SonGoku last updated on 17/Jul/24

You are fantastic! I learn a lot from your elegantt soluions. Thank you very much.

$$\mathrm{You}\:\mathrm{are}\:\mathrm{fantastic}! \\ $$$$\mathrm{I}\:\mathrm{learn}\:\mathrm{a}\:\mathrm{lot}\:\mathrm{from}\:\mathrm{your}\:\mathrm{elegantt}\:\mathrm{soluions}. \\ $$$$\mathrm{Thank}\:\mathrm{you}\:\mathrm{very}\:\mathrm{much}. \\ $$

Commented by SonGoku last updated on 17/Jul/24

Did you use the law of sines? (y/(sin 17°0′50′′)) = ((50)/(sin 90°))

$$\mathrm{Did}\:\mathrm{you}\:\mathrm{use}\:\mathrm{the}\:\mathrm{law}\:\mathrm{of}\:\mathrm{sines}? \\ $$$$\: \\ $$$$\frac{\mathrm{y}}{\mathrm{sin}\:\mathrm{17}°\mathrm{0}'\mathrm{50}''}\:=\:\frac{\mathrm{50}}{\mathrm{sin}\:\mathrm{90}°} \\ $$

Commented by mr W last updated on 17/Jul/24

y=width of river y=50 sin α cos α=((2.9^2 +1.3^2 −1.7^2 )/(2×2.9×1.3))=0.956233 ⇒sin α=(√(1−0.956233^2 ))=0.292605 ⇒y=50×0.292605=14.63m

$${y}={width}\:{of}\:{river} \\ $$$${y}=\mathrm{50}\:\mathrm{sin}\:\alpha \\ $$$$\mathrm{cos}\:\alpha=\frac{\mathrm{2}.\mathrm{9}^{\mathrm{2}} +\mathrm{1}.\mathrm{3}^{\mathrm{2}} −\mathrm{1}.\mathrm{7}^{\mathrm{2}} }{\mathrm{2}×\mathrm{2}.\mathrm{9}×\mathrm{1}.\mathrm{3}}=\mathrm{0}.\mathrm{956233} \\ $$$$\Rightarrow\mathrm{sin}\:\alpha=\sqrt{\mathrm{1}−\mathrm{0}.\mathrm{956233}^{\mathrm{2}} }=\mathrm{0}.\mathrm{292605} \\ $$$$\Rightarrow{y}=\mathrm{50}×\mathrm{0}.\mathrm{292605}=\mathrm{14}.\mathrm{63}{m} \\ $$

Commented by mr W last updated on 17/Jul/24

in the right triangle: (y/(50))=sin α y=50×sin α

$${in}\:{the}\:{right}\:{triangle}: \\ $$$$\frac{{y}}{\mathrm{50}}=\mathrm{sin}\:\alpha \\ $$$${y}=\mathrm{50}×\mathrm{sin}\:\alpha \\ $$

Commented by mr W last updated on 17/Jul/24

Commented by SonGoku last updated on 17/Jul/24

It is always an honor to learn from you. My sincereh tanks.

$$\mathrm{It}\:\mathrm{is}\:\mathrm{always}\:\mathrm{an}\:\mathrm{honor}\:\mathrm{to}\:\mathrm{learn}\:\mathrm{from}\:\mathrm{you}. \\ $$$$\mathrm{My}\:\mathrm{sincereh}\:\mathrm{tanks}. \\ $$

Commented by mr W last updated on 17/Jul/24

$${thanks}\:{to}\:{you}\:{too}! \\ $$

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