A narrow spotlight is placed on level ground 10 m from a vertical wall

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

Problem

A narrow spotlight is placed on level ground 10 m from a vertical wall. The light can rotate in a vertical plane, sweeping a bright spot up and down the wall. At time

t=0

, the spotlight is aimed horizontally so that the spot is at ground level. The angle

\theta(t)

(in radians) that the light makes with the horizontal increases for

t>0

as the light sweeps upward. Let

y(t)

denote the height (in meters) of the bright spot on the wall above ground level. (a) Derive an explicit equation relating

y

and

\theta

for

0<\theta<\pi/2

. (b) Show that

\dfrac{dy}{dt}=10\sec^{2}\!\theta\,\dfrac{d\theta}{dt}

. (c) When the spot is 6 m above the ground, the spotlight is rotating at a constant rate of

0.40\text{ rad/s}

. Calculate the speed (in m/s) at which the spot is moving upward on the wall at that instant. (d) At the instant the spot reaches 6 m, engineers instead wish the spot to rise at a constant speed of

2.0\text{ m/s}

. Determine the angular speed (in rad/s) the spotlight must have at that moment.

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