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Question Number 40875 by Tawa1 last updated on 28/Jul/18

$$2\mathrm{a}\sin \left(\frac{25}{\mathrm{a}}\right)-51=0,\mathrm{find}\mathrm{a}$$

Answered by MrW3 last updated on 29/Jul/18

$${let}^{\prime }shavealookatthefunctionf\left(x\right)=\frac{\sin \left(x\right)}{x}.$$$$weknowfollowingaboutit:$$$$f(-x)=f\left(x\right)\Rightarrow symmetric$$$$\underset{x\to 0}{\lim }f\left(x\right)=1$$$$\underset{x\to \mathrm{\infty }}{\lim }f\left(x\right)=0$$$$max.f\left(x\right)=1$$$$min.f\left(x\right)=-0.2172$$$$i.e.-0.2172⩽f\left(x\right)<1$$$$$$$$2a\sin \left(\frac{25}{a}\right)=50\times \frac{\sin \left(\frac{25}{a}\right)}{\left(\frac{25}{a}\right)}=50\frac{\sin \left(x\right)}{x}=50f\left(x\right)withx=\frac{25}{a}$$$$sincef\left(x\right)<1$$$$\Rightarrow 2a\sin \left(\frac{25}{a}\right)=50f\left(x\right)<50$$$$\Rightarrow 2a\sin \left(\frac{25}{a}\right)-51<-1$$$${that}^{\prime }ttosay2\mathit{a}\mathbf{sin}\left(\frac{25}{\mathit{a}}\right)-51=0isnot$$$$possible.$$$$$$$$\Rightarrow thereisnosolutionforawhich$$$$fulfills2\mathit{a}\mathbf{sin}\left(\frac{25}{\mathit{a}}\right)-51=0.$$

Commented by Tawa1 last updated on 29/Jul/18

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

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