In a laboratory experiment

MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation

Problem

In a laboratory experiment, a narrow laser beam passes from air into a liquid whose index of refraction

n(t)

varies slowly with temperature over time. At the moment the beam strikes the interface, the angle of refraction

r(t)

(in radians) inside the liquid satisfies Snell’s law in the form

r(t)=\arcsin\!\left(\frac{1}{n(t)}\right),

where the index of refraction of air is taken to be 1. At

t=10

s, the liquid’s index of refraction is

n=1.45

and is increasing at a rate of

0.002\,\text{s}^{-1}

. (a) Derive an expression for

\dfrac{dr}{dt}

in terms of

n

and

\dfrac{dn}{dt}

. Do not substitute numerical values yet. (b) Compute

\dfrac{dr}{dt}

t=10

s. Give your answer to five significant figures (units: rad s

^{-1}

). (c) Assuming

\dfrac{dn}{dt}

remains constant, estimate the change in the angle of refraction during the next 3 seconds. State any approximations you use.