The electrical resistance R of a wire varies directly as its length L and inversely as the… — JAMB UTME

The electrical resistance $R$ of a wire varies directly as its length $L$ and inversely as the square of its diameter $d$ . If a wire of length $36\text{ m}$ and diameter $3\text{ mm}$ has a resistance of $16\ \Omega$ , what is the resistance, in ohms, of a wire of the same material with a length of $45\text{ m}$ and a diameter of $5\text{ mm}$ ?

Answer: 7.2 ohms

Answer

The resistance of the wire is 7.2 ohms.

The equation governing the relation is

R = \frac{kL}{d^2}

. Substituting the initial parameters

R=16\ \Omega

L=36\text{ m}

, and

d=3\text{ mm}

gives

16 = \frac{36k}{9} = 4k

, which yields

k = 4

. Using

k = 4

with the new dimensions

L=45\text{ m}

and

d=5\text{ mm}

gives

R = \frac{4 \times 45}{5^2} = \frac{180}{25} = 7.2\ \Omega

Step-by-Step Solution

Formulate the joint variation equation

R = \frac{kL}{d^2}

Direct variation places length

L

in the numerator and inverse variation of the square of diameter

d

places

d^2

in the denominator.

Determine the variation constant

k

k = 4

Substituting

R = 16

L = 36

, and

d = 3

gives

16 = \frac{36k}{9} \implies 16 = 4k \implies k = 4

Calculate the new resistance

R = 7.2\ \Omega

Substituting

k = 4

L = 45

, and

d = 5

into

R = \frac{kL}{d^2}

yields

R = \frac{4 \times 45}{25} = 7.2

Key Concept

Joint Variation involving direct proportionality and inverse square law

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