A) \[\frac{1}{6}\,\sqrt{\frac{7}{3}}\]
B) \[\frac{\sqrt{5}}{6}\]
C) \[\frac{1}{2}\]
D) \[\frac{1}{3}\,\sqrt{\frac{7}{3}}\]
Correct Answer: A
Solution :
\[y-{{x}^{3/2}}=7\left( x\ge 0 \right)\] \[\frac{dy}{dx}\,=\frac{3}{2}{{x}^{1/2}}\] \[\left( \frac{3}{2}\,\sqrt{x} \right)\left( \frac{7-y}{\frac{1}{2}-x} \right)\,=\,-1\] \[\frac{3}{2}{{x}^{2}}\,=\,\,\frac{1}{2}-x\] \[\left( x+1 \right)\left( 3x-1 \right)=0\] \[x=-1\] (rejected) \[x=\frac{1}{3}\] \[y=7+{{x}^{3/2}}=7+{{\left( \frac{1}{3} \right)}^{3/2}}\] \[{{\ell }_{AB}}=\,\sqrt{{{\left( \frac{1}{2}-\frac{1}{3} \right)}^{2}}+{{\left( \frac{1}{3} \right)}^{3}}}\] \[=\,\,\sqrt{\frac{7}{108}}\,=\,\frac{1}{6}\sqrt{\frac{7}{3}}\]You need to login to perform this action.
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