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Question Number 71593 by TawaTawa last updated on 17/Oct/19

Answered by mr W last updated on 17/Oct/19

Commented by mr W last updated on 17/Oct/19

y=a cosh (x/a) s=a sinh (x/a) a+30=a cosh (b/a) 40=a sinh (b/a) (a+30)^2 −40^2 =a^2 60a+30^2 −40^2 ⇒a=((40^2 −30^2 )/(60))=((70)/6) sinh (b/a)=((40)/a)=((24)/7) ⇒b=((70)/6) sinh^(−1) ((24)/7) ⇒?=2b=((70)/3) sinh^(−1) ((24)/7)=45.4 generally: b=((sinh^(−1) ((2sh)/(s^2 −h^2 )))/((2sh)/(s^2 −h^2 )))×s

$${y}={a}\:\mathrm{cosh}\:\frac{{x}}{{a}} \\ $$$${s}={a}\:\mathrm{sinh}\:\frac{{x}}{{a}} \\ $$$${a}+\mathrm{30}={a}\:\mathrm{cosh}\:\frac{{b}}{{a}} \\ $$$$\mathrm{40}={a}\:\mathrm{sinh}\:\frac{{b}}{{a}} \\ $$$$\left({a}+\mathrm{30}\right)^{\mathrm{2}} −\mathrm{40}^{\mathrm{2}} ={a}^{\mathrm{2}} \\ $$$$\mathrm{60}{a}+\mathrm{30}^{\mathrm{2}} −\mathrm{40}^{\mathrm{2}} \\ $$$$\Rightarrow{a}=\frac{\mathrm{40}^{\mathrm{2}} −\mathrm{30}^{\mathrm{2}} }{\mathrm{60}}=\frac{\mathrm{70}}{\mathrm{6}} \\ $$$$\mathrm{sinh}\:\frac{{b}}{{a}}=\frac{\mathrm{40}}{{a}}=\frac{\mathrm{24}}{\mathrm{7}} \\ $$$$\Rightarrow{b}=\frac{\mathrm{70}}{\mathrm{6}}\:\mathrm{sinh}^{−\mathrm{1}} \:\frac{\mathrm{24}}{\mathrm{7}} \\ $$$$\Rightarrow?=\mathrm{2}{b}=\frac{\mathrm{70}}{\mathrm{3}}\:\mathrm{sinh}^{−\mathrm{1}} \:\frac{\mathrm{24}}{\mathrm{7}}=\mathrm{45}.\mathrm{4} \\ $$$$ \\ $$$${generally}: \\ $$$${b}=\frac{\mathrm{sinh}^{−\mathrm{1}} \:\frac{\mathrm{2}{sh}}{{s}^{\mathrm{2}} −{h}^{\mathrm{2}} }}{\frac{\mathrm{2}{sh}}{{s}^{\mathrm{2}} −{h}^{\mathrm{2}} }}×{s} \\ $$

Commented by TawaTawa last updated on 17/Oct/19

God bless you sir. Thanks for your time

$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir}.\:\mathrm{Thanks}\:\mathrm{for}\:\mathrm{your}\:\mathrm{time} \\ $$

Commented by TawaTawa last updated on 17/Oct/19

Thanks sir. God bless you

$$\mathrm{Thanks}\:\mathrm{sir}.\:\mathrm{God}\:\mathrm{bless}\:\mathrm{you} \\ $$

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