A spotlight placed on level ground is 24 ,m south of a vertical warehouse wall

MATHEMATICS · Multi-part · CBSE · Mathematics · Applications of Derivatives · Applications of Derivatives

Problem

A spotlight placed on level ground is 24\,m south of a vertical warehouse wall. The light turns so that its beam sweeps upward along the wall. Let

\theta(t)

denote the angle, in radians, between the beam and the ground. At the instant when

\theta=\dfrac{\pi}{6}

, the light is rotating at a constant rate of

0.20

\,rad\,s

^{-1}

. Ignore any vertical offset of the light above the ground. (a) Derive a formula that expresses the height

y

(in metres) of the illuminated point on the wall in terms of

\theta

. (b) Compute the vertical velocity

\dfrac{dy}{dt}

of the light spot on the wall at the specified instant. (c) Determine the vertical acceleration

\dfrac{d^{2}y}{dt^{2}}

of the spot at the same instant, showing all steps in your reasoning. (d) Interpret the sign of

\dfrac{d^{2}y}{dt^{2}}

in the physical context of the problem.