A power supply delivers a constant electrical power of to a load whose resistance (in…

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

Problem

A power supply delivers a constant electrical power of

150\text{ W}

to a load whose resistance

R

(in ohms) increases over time due to heating. At time

t

, the voltage across the load is

V(t)

volts and the current through it is

I(t)

amperes. Power is related to voltage and current by

P=IV

, and Ohm’s law gives

V=IR

. Suppose that when

t=25\text{ s}

, the resistance is

12\,\Omega

and is increasing at a rate of

0.040\,\Omega/\text{s}

. (1) Express

I

solely in terms of

R

using the given information. (2) Derive a formula for

\dfrac{dI}{dt}

in terms of

R

and

\dfrac{dR}{dt}

. (3) Compute

I

and

\dfrac{dI}{dt}

t=25\text{ s}

. Include appropriate units. (4) The wires connecting the power supply to the load can safely carry current magnitudes up to

4.5\text{ A}

. Assuming

\dfrac{dI}{dt}

remains approximately constant for several seconds, estimate how many additional seconds after

t=25\text{ s}

the current will exceed the safe limit. State any assumption you make and justify your reasoning.