A helicopter is flying along the curve given | vedclass

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(C) Let the point on the curve be $P(x, x^{3/2} + 7)$. The soldier is at $A(1/2, 7)$.The squared distance $D^2 = (x - 1/2)^2 + (x^{3/2} + 7 - 7)^2 = (

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$\frac{1}{6}\sqrt{\frac{7}{3}}$

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(C) Let the point on the curve be $P(x, x^{3/2} + 7)$. The soldier is at $A(1/2, 7)$.The squared distance $D^2 = (x - 1/2)^2 + (x^{3/2} + 7 - 7)^2 = (x - 1/2)^2 + x^3$.Let $f(x) = (x - 1/2)^2 + x^3$. For minimum distance,$f'(x) = 0$.$f'(x) = 2(x - 1/2) + 3x^2 = 3x^2 + 2x - 1 = 0$.$(3x - 1)(x + 1) = 0$. Since $x \geq 0$,we have $x = 1/3$.The point $P$ is $(1/3, (1/3)^{3/2} + 7) = (1/3, 7 + \frac{1}{3\sqrt{3}})$.The distance $AD = \sqrt{(1/3 - 1/2)^2 + (7 + \frac{1}{3\sqrt{3}} - 7)^2} = \sqrt{(-1/6)^2 + (\frac{1}{3\sqrt{3}})^2}$.$AD = \sqrt{\frac{1}{36} + \frac{1}{27}} = \sqrt{\frac{3 + 4}{108}} = \sqrt{\frac{7}{108}} = \frac{1}{6}\sqrt{\frac{7}{3}}$.

$A$ helicopter is flying along the curve given by $y = x^{3/2} + 7, (x \geq 0)$. $A$ soldier positioned at the point $(1/2, 7)$ wants to shoot down the helicopter when it is nearest to him. Then this nearest distance is

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