Question

The resistance R of wire of fixed length is related to the diameter x by an inverse square law, that is, by a function of the form \(R\left( x \right) = k{x^{ - {\bf{2}}}}\).

(a) A wire of fixed length and 0.005 meters in diameter has a resistance of 140 ohms. Find the value of k.

(b) Find the resistance of a wire made of the same material and of the same length as the wire in part (a) but with a diameter of 0.008 meters.

Short answer

a. \(k = 0.0035\)

b. 54.7 ohms

Step-by-step solution

Find the value of k

Substitute 0.005 for x in the equation \(R\left( x \right) = k{x^{ - 2}}\).

\(\begin{aligned}R\left( x \right) &= k{x^{ - 2}}\\140 &= k{\left( {0.005} \right)^{ - 2}}\\k &= 140{\left( {0.005} \right)^2}\\ &= 0.0035\end{aligned}\)

Express the pressure as a function of depth 

The resistance of the wire as a function of length is

\(R\left( x \right) = 0.0035{x^2}\).

Substitute 0.008 for x in the function \(R\left( x \right) = 0.0035{x^2}\).

\(\begin{aligned}R\left( {0.008} \right) = 0.0035{\left( {0.008} \right)^{ - 2}}\\ \approx 54.7\;{\rm{ohms}}\end{aligned}\)

So, the value of k is 0.0035, and the resistance of the 0.008-meter-long wire is 54.7 ohms.