A helicopter is flying along the curve given by y -- x3/2 = 7, (x [ ] 0).

Question

A helicopter is flying along the curve given by y -- x3/2 = 7, (x $$ \ge $$ 0). A soldier positioned at the point $$\left( {{1 \over 2},7} \right)$$ wants to shoot down the helicopter when it is nearest to him. Then this nearest distance is -

Accepted Answer

$$y - {x^{3/2}} = 7\left( {x \ge 0} \right)$$

$${{dy} \over {dx}} = {3 \over 2}{x^{1/2}}$$

$$\left( {{3 \over 2}\sqrt x } \right)\left( {{{7 - y} \over {{1 \over 2} - x}}} \right) =  - 1$$

$$\left( {{3 \over 2}\sqrt x } \right)\left( {{{ - {x^{3/2}}} \over {{1 \over 2} - x}}} \right) =  - 1$$

$${3 \over 2}.{x^2} = {1 \over 2} - x$$

$3{x^2} = 1 - 2x$

$3{x^2} + 2x - 1 = 0$

$3{x^2} + 3x - x - 1 = 0$

$$\left( {x + 1} \right)\left( {3x - 1} \right) = 0$$

$$ \therefore $$  $x =  - 1$  (rejected)

$$x = {1 \over 3}$$

$$y = 7 + {x^{3/2}} = 7 + {\left( {{1 \over 3}} \right)^{3/2}}$$

$${\ell _{AB}} = \sqrt {{{\left( {{1 \over 2} - {1 \over 3}} \right)}^2} + {{\left( {{1 \over 3}} \right)}^3}}  = \sqrt {{1 \over {36}} + {1 \over {27}}} $$

$$ = \sqrt {{{3 + 4} \over {9 \times 12}}} $$

$$ = \sqrt {{7 \over {108}}}  = {1 \over 6}\sqrt {{7 \over 3}} $$

Solutions · 1