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
$${y}={a}\:\mathrm{cosh}\:\frac{{x}}{{a}} \\ $$$${s}={a}\:\mathrm{sinh}\:\frac{{x}}{{a}} \\ $$$${a}+\mathrm{30}={a}\:\mathrm{cosh}\:\frac{{b}}{{a}} \\ $$$$\mathrm{40}={a}\:\mathrm{sinh}\:\frac{{b}}{{a}} \\ $$$$\left({a}+\mathrm{30}\right)^{\mathrm{2}} −\mathrm{40}^{\mathrm{2}} ={a}^{\mathrm{2}} \\ $$$$\mathrm{60}{a}+\mathrm{30}^{\mathrm{2}} −\mathrm{40}^{\mathrm{2}} \\ $$$$\Rightarrow{a}=\frac{\mathrm{40}^{\mathrm{2}} −\mathrm{30}^{\mathrm{2}} }{\mathrm{60}}=\frac{\mathrm{70}}{\mathrm{6}} \\ $$$$\mathrm{sinh}\:\frac{{b}}{{a}}=\frac{\mathrm{40}}{{a}}=\frac{\mathrm{24}}{\mathrm{7}} \\ $$$$\Rightarrow{b}=\frac{\mathrm{70}}{\mathrm{6}}\:\mathrm{sinh}^{−\mathrm{1}} \:\frac{\mathrm{24}}{\mathrm{7}} \\ $$$$\Rightarrow?=\mathrm{2}{b}=\frac{\mathrm{70}}{\mathrm{3}}\:\mathrm{sinh}^{−\mathrm{1}} \:\frac{\mathrm{24}}{\mathrm{7}}=\mathrm{45}.\mathrm{4} \\ $$$$ \\ $$$${generally}: \\ $$$${b}=\frac{\mathrm{sinh}^{−\mathrm{1}} \:\frac{\mathrm{2}{sh}}{{s}^{\mathrm{2}} −{h}^{\mathrm{2}} }}{\frac{\mathrm{2}{sh}}{{s}^{\mathrm{2}} −{h}^{\mathrm{2}} }}×{s} \\ $$
Commented by TawaTawa last updated on 17/Oct/19
$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir}.\:\mathrm{Thanks}\:\mathrm{for}\:\mathrm{your}\:\mathrm{time} \\ $$
Commented by TawaTawa last updated on 17/Oct/19
$$\mathrm{Thanks}\:\mathrm{sir}.\:\mathrm{God}\:\mathrm{bless}\:\mathrm{you} \\ $$