A portable work light is mounted on a vertical lift directly above a point on flat ground

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

Problem

A portable work light is mounted on a vertical lift directly above a point on flat ground. The lamp is being raised so that its height

h(t)

(in meters) increases at a constant rate, i.e.

\frac{dh}{dt} = 0.20\,\text{m/s}

. A technician who is

1.9\,\text{m}

tall stands

4.0\,\text{m}

from the base of the lift. Let

y(t)

denote the length (in meters) of the technician’s shadow on the ground at time

t

seconds, and assume all motion is confined to a single vertical plane. (1) Part 1: (a) Express

y

in terms of

h

. Show all reasoning that leads to your formula. (2) Part 2: (b) Differentiate your relation implicitly with respect to time to obtain an expression for

\frac{dy}{dt}

in terms of

h

and

\frac{dh}{dt}

. (3) Part 3: (c) Calculate

\frac{dy}{dt}

when

h = 8.0\,\text{m}

. Include correct units in your answer. (4) Part 4: (d) State whether the technician’s shadow is lengthening or shortening at that instant and explain how the sign of

\frac{dy}{dt}

supports your conclusion.

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