# Question #9a735

Jan 15, 2018

The plane's distance is increasing at $330.72$ miles/hour
when the plane is at
$4$ miles from me.

#### Explanation:

Let $y = 3$ mile altitude at which plane is flying and $y$ is

constant. $z = 4$ mile distance from me. Let $x$ be the horizontal

distance covered from iniitial position at my overhead.

${x}^{2} + {y}^{2} = {z}^{2} \mathmr{and} {x}^{2} + {3}^{2} = {4}^{2} \therefore {x}^{2} = 16 - 9$ or

$x = \sqrt{7}$ mile $\frac{\mathrm{dx}}{\mathrm{dt}} = x ' = 500$ mph.

${x}^{2} + {y}^{2} = {z}^{2}$ . Differentiating both sides w.r.t time $t$

we get $2 x \cdot x ' + 2 y \cdot y ' = 2 z \cdot z ' \mathmr{and} x \cdot x ' + y \cdot y ' = z \cdot z '$

$y ' = 0$ since $y =$constant $\therefore x \cdot x ' + y \cdot 0 = z \cdot z '$ or

$\therefore x \cdot x ' = z \cdot z ' \mathmr{and} \sqrt{7} \cdot 500 = 4 \cdot z '$ or

$z ' = \frac{\sqrt{7} \cdot 500}{4} \approx 330.72$ miles/hour

The plane's distance is increasing at $330.72$ miles/hour

when the plane is at $4$ miles from me. [Ans]