Question Number 200366 by ajfour last updated on 17/Nov/23
Commented by ajfour last updated on 17/Nov/23
$${The}\:{red}\:{circular}\:{arc}\:{length}\:{is}\:{equal} \\ $$$$\:{to}\:{blue}\:{arclength}\:\:{of}\:{parabola}.\:{Find}\: \\ $$$${the}\:{equation}\:{of}\:{parabola}\:{in}\:{the}\:{form}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{y}={ax}^{\mathrm{2}} +{c}. \\ $$
Answered by Frix last updated on 17/Nov/23
$$\mathrm{Because}\:\mathrm{of}\:\mathrm{the}\:\mathrm{intersections}\:\Rightarrow\:{c}=\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}−\frac{{a}}{\mathrm{4}} \\ $$$$\underset{\mathrm{0}} {\overset{\frac{\mathrm{1}}{\mathrm{2}}} {\int}}\sqrt{\left(\frac{{dy}}{{dx}}\right)^{\mathrm{2}} +\mathrm{1}}\:{dx}=\frac{\pi}{\mathrm{6}} \\ $$$$\frac{\sqrt{{a}^{\mathrm{2}} +\mathrm{1}}}{\mathrm{4}}+\frac{\mathrm{ln}\:\left({a}+\sqrt{{a}^{\mathrm{2}} +\mathrm{1}}\right)}{\mathrm{4}{a}}=\frac{\pi}{\mathrm{6}} \\ $$$${a}\approx.\mathrm{543171783} \\ $$$${c}\approx.\mathrm{730232458} \\ $$
Commented by ajfour last updated on 17/Nov/23
$${Thanks}!\: \\ $$