An electrical test bench delivers a constant 80\text{ V} direct-current supply to four sep… — Armed Services Vocational Aptitude Battery (ASVAB)

An electrical test bench delivers a constant $80\text{ V}$ direct-current supply to four separate parallel circuit arrangements. Match each circuit configuration on the left with its corresponding equivalent resistance ( $R_{eq}$ ) and total supply current ( $I_T$ ) on the right.

Two resistors of $40\ \Omega$ and $10\ \Omega$ connected in parallel across the supply $R_{eq} = 8\ \Omega$ and $I_T = 10\text{ A}$
Four identical $64\ \Omega$ resistive heating elements connected in parallel $R_{eq} = 16\ \Omega$ and $I_T = 5\text{ A}$
Three parallel branches containing $120\ \Omega$ , $60\ \Omega$ , and $40\ \Omega$ resistors $R_{eq} = 20\ \Omega$ and $I_T = 4\text{ A}$
Two parallel branches containing a $150\ \Omega$ sensor and a $75\ \Omega$ resistor $R_{eq} = 50\ \Omega$ and $I_T = 1.6\text{ A}$

Answer

The configuration with

40\ \Omega

and

10\ \Omega

matches

R_{eq} = 8\ \Omega, I_T = 10\text{ A}

; four

64\ \Omega

elements match

R_{eq} = 16\ \Omega, I_T = 5\text{ A}

; three branches of

120\ \Omega, 60\ \Omega, 40\ \Omega

match

R_{eq} = 20\ \Omega, I_T = 4\text{ A}

; and two branches of

150\ \Omega, 75\ \Omega

match

R_{eq} = 50\ \Omega, I_T = 1.6\text{ A}

In any direct-current parallel network, the total equivalent resistance (

R_{eq}

) is calculated using the reciprocal formula

\frac{1}{R_{eq}} = \sum \frac{1}{R_n}

(or

R/N

for identical branches). Total current drawn from the supply is determined by Ohm's Law:

I_T = \frac{V_S}{R_{eq}}

. Evaluating each configuration against the constant

80\text{ V}

bus correctly establishes all four resistance and current pairs.

Step-by-Step Solution

Calculate the equivalent resistance and total current for the

40\ \Omega

and

10\ \Omega

parallel branch pair.

R_{eq} = \frac{40 \times 10}{40 + 10} = 8\ \Omega

;

I_T = \frac{80\text{ V}}{8\ \Omega} = 10\text{ A}

Two unequal parallel resistors follow the product-over-sum rule, and total current is determined by Ohm's Law (

I = V/R

Calculate the equivalent resistance and current for the four parallel

64\ \Omega

elements.

R_{eq} = \frac{64}{4} = 16\ \Omega

;

I_T = \frac{80\text{ V}}{16\ \Omega} = 5\text{ A}

Identical parallel resistors simplify to branch resistance divided by the number of branches (

R/N

Calculate the equivalent resistance and current for the three parallel branches (

120\ \Omega, 60\ \Omega, 40\ \Omega

\frac{1}{R_{eq}} = \frac{1}{120} + \frac{2}{120} + \frac{3}{120} = \frac{6}{120} = \frac{1}{20} \implies R_{eq} = 20\ \Omega

;

I_T = \frac{80\text{ V}}{20\ \Omega} = 4\text{ A}

Three or more unequal parallel branches require finding the reciprocal sum over a common denominator.

Calculate the equivalent resistance and current for the

150\ \Omega

and

75\ \Omega

parallel branches.

R_{eq} = \frac{150 \times 75}{150 + 75} = \frac{11250}{225} = 50\ \Omega

;

I_T = \frac{80\text{ V}}{50\ \Omega} = 1.6\text{ A}

Applying product-over-sum and dividing bus voltage by equivalent resistance yields the total circuit current.

Key Concept

Parallel circuit rules require that full bus voltage appears across each branch, equivalent resistance is always lower than the smallest branch resistance, and total current equals the sum of all individual branch currents.

Estimated Time:2m 0s

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