A cylindrical copper wire has constant volume

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

Problem

A cylindrical copper wire has constant volume V=4.0 cm3V=4.0\text{ cm}^3. Its electrical resistivity ρ=1.7×10−6 Ω⋅cm\rho=1.7\times 10^{-6}\,\Omega\cdot\text{cm} is constant. The resistance of the wire is given by R=ρLAR=\dfrac{\rho L}{A}, where L(t)L(t) is the length in centimeters and A(t)A(t) is the cross-sectional area in square centimeters. The wire is being stretched so that its length is increasing at dLdt=0.060 cm/s\dfrac{dL}{dt}=0.060\text{ cm/s} when L=20 cmL=20\text{ cm}. (1) Using the volume constraint, write AA as a function of LL. Simplify your expression. (2) Derive an expression for dRdt\dfrac{dR}{dt} in terms of LL, dLdt\dfrac{dL}{dt}, and known constants. Show every differentiation step. (3) Evaluate dRdt\dfrac{dR}{dt} at the instant when L=20 cmL=20\text{ cm} and dLdt=0.060 cm/s\dfrac{dL}{dt}=0.060\text{ cm/s}. Give your answer in ohms per second (Ω/s\Omega/\text{s}) to an appropriate number of significant figures. (4) Explain, in practical terms, what your result implies about how the stretching process affects the wire’s ability to conduct electric current.

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