A mobile radar tracking unit contains two heating elements connected in parallel across a … — Armed Services Vocational Aptitude Battery (ASVAB)

A mobile radar tracking unit contains two heating elements connected in parallel across a common direct-current voltage rail. The primary heating element has a resistance of $18\ \Omega$ , and the secondary heating element has a resistance of $9\ \Omega$ . What is the total equivalent resistance of this circuit?

A
$27\ \Omega$
$6\ \Omega$ Answer
C
$13.5\ \Omega$
D
$2\ \Omega$

Answer

The total equivalent resistance of the parallel circuit is

6\ \Omega

The equivalent resistance of parallel resistors is found using the formula

\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2}

or the shortcut

R_{eq} = \frac{R_1 \times R_2}{R_1 + R_2}

. Substituting

18\ \Omega

and

9\ \Omega

yields

\frac{18 \times 9}{18 + 9} = \frac{162}{27} = 6\ \Omega

, which is properly less than the smallest branch resistance (

9\ \Omega

Step-by-Step Solution

Identify the standard equivalent resistance relationship for two unequal parallel branches

\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2}

R_{eq} = \frac{R_1 \times R_2}{R_1 + R_2}

Parallel branches provide multiple paths for current, resulting in a total equivalent resistance that is always smaller than the lowest individual branch resistance.

Substitute the given resistances (

R_1 = 18\ \Omega

and

R_2 = 9\ \Omega

) into the product-over-sum formula

R_{eq} = \frac{18 \times 9}{18 + 9} = \frac{162}{27}

Using the product-over-sum method allows direct calculation without converting to fractions with a common denominator.

Divide the product by the sum to determine the equivalent resistance

R_{eq} = 6\ \Omega

Simplifying

\frac{162}{27}

yields the total opposition to current offered by the two parallel elements.

Key Concept

Total Equivalent Resistance of Unequal Parallel Branches

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