A portable spotlight is placed on level ground 2 m from the base of a vertical wall

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

Problem

A portable spotlight is placed on level ground 2 m from the base of a vertical wall. The beam makes an angle

\theta

with the ground and illuminates a point

P

located

y

meters above the ground on the wall. At the instant when

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

, the spotlight is rotating so that

\dfrac{d\theta}{dt}=0.20

rad/s. (a) Express

y

in terms of

\theta

. (b) Obtain an equation relating

\dfrac{dy}{dt}

\dfrac{d\theta}{dt}

at any instant. (c) Calculate the speed (in m/s) at which the illuminated point

P

is moving upward on the wall when

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

. (d) State the units of each quantity in part (b) and briefly interpret the physical meaning of the sign of

\dfrac{dy}{dt}

in this context.