All practice questions

Determine the missing constants in the completely factored form of the given polynomial.

Fill in the blanks below

The polynomial

5x^2 - 5x - 60

can be factored completely. When written in the form

a(x - b)(x + c)

, where

b

and

c

are positive integers, the value of

a

is , the value of

b

is , and the value of

c

is .

Show answer & explanation

Answer

The constants are a = 5, b = 4, and c = 3.

The correct values are found by first factoring out the greatest common factor of

5

, leaving the quadratic expression

x^2 - x - 12

. This quadratic expression factors into two binomials by finding two numbers that multiply to

-12

and add to

-1

, which are

-4

and

3

. This yields the fully factored form

5(x - 4)(x + 3)

. By matching this result to the required algebraic structure

a(x - b)(x + c)

, we can directly identify that

a = 5

,

b = 4

, and

c = 3

.

Step-by-Step Solution

1

Find the greatest common factor (GCF) of the terms in the polynomial

5x^2 - 5x - 60

.

The GCF is

5

. Factoring it out gives

5(x^2 - x - 12)

.

Extracting the GCF simplifies the trinomial and makes it easier to factor into binomials.

2

Factor the remaining quadratic trinomial

x^2 - x - 12

.

We look for two integers that multiply to

-12

and add to

-1

.

This is the standard procedure for factoring a monic quadratic trinomial of the form

x^2 + bx + c

.

3

Identify the proper integers and write the binomial factors.

The integers are

-4

and

3

. The factored trinomial is

(x - 4)(x + 3)

.

(-4) \times 3 = -12

and

-4 + 3 = -1

.

4

Combine the factors and match them to the target form

a(x - b)(x + c)

.

The completely factored expression is

5(x - 4)(x + 3)

. Matching this to

a(x - b)(x + c)

yields

a = 5

,

b = 4

, and

c = 3

.

This establishes the precise values corresponding to the blanks in the question.

Key Concept

Factoring out a greatest common factor followed by factoring a quadratic trinomial.

Question 22Question →

The area of a rectangular solar panel is modeled by the algebraic expression $w^2 - 5w + 6$ . Which of the following represents the completely factored form of this expression?

(w - 2)(w - 3)

(w + 2)(w + 3)

(w - 6)(w + 1)

(w + 6)(w - 1)

Show answer & explanation

Answer: $(w - 2)(w - 3)$

Answer

The completely factored form of the expression is

(w - 2)(w - 3)

.

To factor the quadratic expression

w^2 - 5w + 6

, you must find two integers that multiply to the constant term (6) and add to the middle term's coefficient (-5). The integers -2 and -3 satisfy both conditions because

(-2) \times (-3) = 6

and

(-2) + (-3) = -5

. Placing these into binomials gives the correct factored form.

Step-by-Step Solution

1

Identify the type of expression

The expression is a quadratic trinomial in the form

w^2 + bw + c

, where

b = -5

and

c = 6

.

Recognizing the structure helps determine the factoring method.

2

Find the factors of the constant term

The factors of 6 are (1, 6), (-1, -6), (2, 3), and (-2, -3).

The binomial factors must contain two numbers that multiply to the constant term.

3

Select the factor pair that adds to the middle coefficient

The sum of -2 and -3 is -5.

The two numbers must add up to the coefficient of the middle term.

4

Write the final factored expression

(w - 2)(w - 3)

Substituting the correct numbers into the binomial format

(w + p)(w + q)

.

Key Concept

Factoring Monic Quadratic Trinomials

Question 23Question →

A flight line supervisor is dividing $480$ hours of scheduled maintenance tasks between two mechanic crews in a ratio of $3:5$ . If the larger share of hours is assigned to the senior crew, how many maintenance hours does the senior crew receive?

160

180

288

300

Show answer & explanation

Answer: 300

Answer

The senior crew receives 300 maintenance hours.

To solve the problem, first find the total number of parts in the ratio:

3 + 5 = 8

parts. Next, divide the total number of hours by the total parts to find the value of one part:

480 \div 8 = 60

hours. Finally, multiply the value of one part by

5

(the larger share) to find the senior crew's allocation:

60 \times 5 = 300

hours.

Step-by-Step Solution

1

Find the total number of parts in the ratio by adding the terms together.

3 + 5 = 8

total parts.

To distribute a total amount according to a ratio, you must first determine how many equal parts make up the whole.

2

Determine the value of a single part by dividing the total hours by the total number of parts.

480 \div 8 = 60

hours per part.

This establishes the baseline unit value that corresponds to '1' in the ratio.

3

Multiply the value of a single part by the larger number in the ratio to find the senior crew's share.

60 \times 5 = 300

hours.

The problem states the senior crew receives the larger share, which corresponds to the '5' in the

3:5

ratio.

Key Concept

To divide a total quantity into a given ratio, add the terms to find the total number of parts, divide the total quantity by this sum to find the value of one part, and then multiply by the desired ratio term.

Estimated Time:1m 0s

Question 24Question →

Match each military travel scenario with its calculated average speed or travel time.

Click a left item, then click its matching right item

Show answer & explanation

Answer

The matches are: convoy speed is 45 mph, patrol boat time is 30 minutes, helicopter speed is 160 mph, and humvee time is 120 minutes.

Each scenario requires applying the fundamental relationship between distance, rate, and time (

d = rt

). For scenarios asking for minutes, the calculated hours are multiplied by 60. For scenarios given in minutes but requiring mph, the minutes are divided by 60.

Step-by-Step Solution

1

Calculate the speed for the convoy.

180 \text{ miles} \div 4 \text{ hours} = 45 \text{ mph}

Average speed is found by dividing total distance by total time.

2

Calculate the travel time for the patrol boat.

15 \text{ miles} \div 30 \text{ mph} = 0.5 \text{ hours}

.

0.5 \times 60 = 30 \text{ minutes}

.

Time is distance divided by rate. The result in hours must be multiplied by 60 to find minutes.

3

Calculate the speed for the helicopter.

90 \text{ minutes} = 1.5 \text{ hours}

.

240 \text{ miles} \div 1.5 \text{ hours} = 160 \text{ mph}

.

Speed in mph requires the time to be in hours, so minutes are divided by 60 first.

4

Calculate the travel time for the humvee.

110 \text{ miles} \div 55 \text{ mph} = 2 \text{ hours}

.

2 \times 60 = 120 \text{ minutes}

.

Divide distance by rate to get hours, then multiply by 60 for minutes.

Key Concept

Calculating Rate and Time with Unit Conversions

Estimated Time:1m 30s

Question 25Question →

Assume

x

and

y

are non-zero real numbers. When the algebraic expression

\frac{(2x^{-3}y^4)^3}{4x^5y^{-2}}

is fully simplified to the form

cx^ay^b

, what is the integer value of

b

?

Show answer & explanation

Answer: 14

Answer

14

The expression simplifies to

2x^{-14}y^{14}

. Because the question asks for the exponent of

y

, which is denoted by

b

in the target expression

cx^ay^b

, the correct value is 14.

Step-by-Step Solution

1

Apply the power of a product rule to the numerator.

(2x^{-3}y^4)^3 = 2^3(x^{-3})^3(y^4)^3 = 8x^{-9}y^{12}

When an algebraic term with coefficients and variables is raised to a power, the outer exponent must be distributed to each individual factor inside the parentheses.

2

Set up the division by applying the quotient rule for each base.

\frac{8x^{-9}y^{12}}{4x^5y^{-2}} = \left(\frac{8}{4}\right) \cdot \left(\frac{x^{-9}}{x^5}\right) \cdot \left(\frac{y^{12}}{y^{-2}}\right)

Separating the expression by identical bases allows the independent application of the quotient rule for exponents.

3

Evaluate the exponent for the

y

term to find the value of

b

.

y^{12} \div y^{-2} = y^{12 - (-2)} = y^{12 + 2} = y^{14}

When dividing exponential terms with the same base, subtract the denominator's exponent from the numerator's exponent. Care must be taken with the double negative.

4

Identify the requested integer value.

b = 14

The question specifically asks for the integer value of

b

in the final simplified form

cx^ay^b

, which represents the exponent of the

y

term.

Key Concept

Applying the power of a power and quotient of powers rules for variable exponents.

Question 26Question →

Match each polynomial expression on the left with its completely factored equivalent on the right.

Click a left item, then click its matching right item

Show answer & explanation

Answer

The correct matches pair each polynomial with its completely factored form by applying techniques such as factoring out the greatest common factor (GCF), recognizing difference of squares, and factoring trinomials.

Each polynomial is completely factored by extracting any Greatest Common Factors and correctly applying fundamental algebraic patterns, including the difference of squares, perfect square trinomials, and standard quadratic trinomial factoring.

Step-by-Step Solution

1

Analyze the first polynomial:

x^2 - 3x - 10

.

It factors to

(x - 5)(x + 2)

.

This is a basic trinomial where we need factors of -10 that add up to -3.

2

Analyze the second polynomial:

2x^2 - 18

.

It factors to

2(x - 3)(x + 3)

.

Factoring out the GCF of 2 leaves

x^2 - 9

, which is a difference of squares and factors further into

(x - 3)(x + 3)

.

3

Analyze the third polynomial:

3x^2 - 6x

.

It factors to

3x(x - 2)

.

The two terms share a common factor of

3x

. Extracting this GCF directly yields the factored form.

4

Analyze the fourth polynomial:

x^2 - 10x + 25

.

It factors to

(x - 5)^2

.

This matches the perfect square trinomial pattern

a^2 - 2ab + b^2 = (a - b)^2

with

a=x

and

b=5

.

Key Concept

Factoring Polynomials

Question 27Question →

Find the completely factored form of the given polynomial:

2x^2 + 2x - 40

Assume all missing values in the blanks are positive integers.

Fill in the blanks below

The completely factored expression is (x + )(x - ).

Show answer & explanation

Answer

The missing values are 2, 5, and 4. The completely factored expression is 2(x + 5)(x - 4).

Factoring out the greatest common factor of 2 leaves

x^2 + x - 20

. This trinomial factors into

(x + 5)(x - 4)

because 5 and -4 multiply to give -20 and add to give 1. Thus, the complete factored form is

2(x + 5)(x - 4)

, corresponding to the blank values 2, 5, and 4.

Step-by-Step Solution

1

Factor out the greatest common factor (GCF) from all the terms in the polynomial.

2(x^2 + x - 20)

Identifying and extracting the GCF of 2 simplifies the remaining trinomial, making it easier to factor.

2

Factor the resulting quadratic trinomial

x^2 + x - 20

.

Find two numbers that multiply to -20 and add to 1 (the coefficient of x).

To factor a trinomial of the form

x^2 + bx + c

, you must find two factors of

c

that add up to

b

.

3

Identify the two numbers that satisfy the conditions.

The numbers are 5 and -4.

5 \times (-4) = -20

and

5 + (-4) = 1

.

4

Write the complete factorization by combining the GCF and the factored binomials.

2(x + 5)(x - 4)

This represents the original polynomial completely factored into its simplest terms.

Key Concept

Factoring Polynomials with a Greatest Common Factor

Question 28Question →

The total revenue from selling a batch of electronic components is modeled by the polynomial $15x^2 - 14x - 8$ . If the number of units sold is given by the expression $(5x + 2)$ , what binomial represents the price per unit?

Show answer & explanation

Answer: 3x-4; (3x-4); 3x - 4; (3x - 4)

Answer

3x - 4

The correct answer is derived by factoring the revenue polynomial

15x^2 - 14x - 8

. Because revenue equals the number of units multiplied by the price per unit, the missing price expression is the binomial that, when multiplied by

(5x + 2)

, yields the given polynomial. By factoring the expression, we find that

15x^2 - 14x - 8 = (5x + 2)(3x - 4)

, confirming

(3x - 4)

is the correct binomial.

Step-by-Step Solution

1

Set up the algebraic relationship based on the context.

Total Revenue = (Number of Units) × (Price per Unit), which translates to

15x^2 - 14x - 8 = (5x + 2) \times (\text{Price per Unit})

.

This establishes that the price per unit is the missing factor of the given polynomial.

2

Determine the factoring method for the trinomial

15x^2 - 14x - 8

.

Using the ac-method, multiply the leading coefficient (15) by the constant term (-8) to get

-120

.

This is the standard approach for factoring a trinomial where the leading coefficient is greater than 1.

3

Find two integers that multiply to

-120

and add to the middle coefficient,

-14

.

The required integers are

-20

and

6

, because

-20 \times 6 = -120

and

-20 + 6 = -14

.

These integers allow us to correctly split the middle term for factoring by grouping.

4

Rewrite the trinomial and factor by grouping.

15x^2 - 20x + 6x - 8 = 5x(3x - 4) + 2(3x - 4) = (5x + 2)(3x - 4)

.

Grouping extracts the greatest common factor from pairs of terms, revealing the two binomial factors.

5

Identify the missing factor.

Since one factor is given as

(5x + 2)

, the other factor representing the price is

(3x - 4)

.

This completes the solution, satisfying the initial equation.

Key Concept

Factoring Non-Monic Trinomials

Question 29Question →

Factor the trinomial $x^2 + 2x - 15$ . Which of the following choices shows the correct factorization?

(x + 5)(x - 3)

(x - 5)(x + 3)

(x + 5)(x + 3)

(x - 15)(x + 2)

Show answer & explanation

Answer: $(x + 5)(x - 3)$

Answer

The correct factorization is

(x + 5)(x - 3)

.

To factor the quadratic expression

x^2 + 2x - 15

, we need to find two numbers that multiply to

-15

and add up to

2

. The numbers

5

and

-3

meet these criteria (

5 \times -3 = -15

and

5 + -3 = 2

). Therefore, the completely factored form is

(x + 5)(x - 3)

.

Step-by-Step Solution

1

Identify the target sum and product for the factors.

The target sum is the coefficient of the middle term (

2

), and the target product is the constant term (

-15

).

To factor a trinomial

x^2 + bx + c

into

(x + p)(x + q)

, we need to find two integers

p

and

q

that multiply to

c

and add up to

b

.

2

Find the two integers that satisfy both conditions.

The factor pairs of

-15

are

(-1, 15)

,

(1, -15)

,

(-3, 5)

, and

(3, -5)

. The pair that adds to

2

is

-3

and

5

.

Checking the sums, only

5 + (-3) = 2

provides the correct middle coefficient.

3

Write the factored form using the selected integers.

The factored form is

(x + 5)(x - 3)

.

Substituting

5

and

-3

into the binomial framework

(x + p)(x + q)

completes the factorization.

Key Concept

Factoring Monic Quadratic Trinomials

Question 30Question →

Read the compensation scenario below and fill in the blanks with the correct numerical values.

Fill in the blanks below

A salon stylist earns a weekly base salary of

450. She also receives a 20% commission on all chemical treatments and a 10% commission on retail product sales. In a single week, she bills

850 for chemical treatments and sells

240 in retail products. For this week, the stylist earned a total of

in commissions, which brings her total gross pay to $.

Show answer & explanation

Answer

The stylist earned

194 in total commissions, making her total gross pay

644.

The correct commission is found by multiplying each sales category by its respective percentage (

850 \times 0.20 = 170

and

240 \times 0.10 = 24

), summing them (

170 + 24 = 194

), and adding the result to the

450 base salary for a total of

644.

Step-by-Step Solution

1

Calculate the commission earned from chemical treatments.

850 \times 0.20 = 170

The stylist earns 20% commission on the $850 billed for these specific services.

2

Calculate the commission earned from retail product sales.

240 \times 0.10 = 24

The stylist earns 10% commission on the $240 worth of retail sales.

3

Determine the total commission earned for the week.

170 + 24 = 194

Total commission is the sum of the commissions from both categories.

4

Calculate the total gross pay.

450 + 194 = 644

Total gross pay is the base salary plus the total commission earned.

Key Concept

Calculating varied commission rates on different sales categories and combining them with a base salary to find gross earnings.

Estimated Time:1m 30s

Question 31Question →

A hospital administrator places $\$12,000$ into a fixed-term savings certificate that earns an annual simple interest rate of $4.2\%$ . If no other deposits or withdrawals are made, what is the total accumulated balance, in dollars, of the certificate after exactly 30 months?

Show answer & explanation

Answer: 13260

Answer

The total accumulated balance is 13260.

The correct answer is found by first converting the 30-month time period into 2.5 years (

30 \div 12 = 2.5

), and the

4.2\%

interest rate into the decimal

0.042

(

4.2 \div 100 = 0.042

). Using the simple interest formula

I = P \times R \times T

, the interest earned is

\$12,000 \times 0.042 \times 2.5 = \$1,260

. Adding this interest to the original principal of

\$12,000

gives a total accumulated balance of

\$13,260

.

Step-by-Step Solution

1

Convert the interest rate to a decimal and the time period to years.

R = 4.2\% = 0.042

, and

T = 30 \text{ months} \div 12 \text{ months/year} = 2.5 \text{ years}

.

The standard simple interest formula requires the interest rate as a decimal and the time expressed in years.

2

Calculate the total interest earned using the simple interest formula

I = P \times R \times T

.

I = 12000 \times 0.042 \times 2.5 = 1260

This determines how much additional money the certificate earns over the 30-month period.

3

Calculate the total accumulated balance by adding the interest earned to the original principal.

Total Balance =

12000 + 1260 = 13260

The question asks for the total accumulated balance, which includes both the initial deposit (principal) and the earned interest.

Key Concept

Calculating simple interest over a fractional year and determining the total accumulated amount.

Question 32Question →

A mechanic serviced the following number of vehicles over a five-day period: $10$ , $15$ , $4$ , $11$ , and $5$ .

What is the median number of vehicles serviced during this period?

4

9

10

11

Show answer & explanation

Answer: $10$

Answer

The median number of vehicles serviced is

10

.

To find the median, the values must first be ordered from least to greatest:

4

,

5

,

10

,

11

,

15

. The median is the middle number in this ordered list, which is

10

.

Step-by-Step Solution

1

Arrange the numbers in ascending order.

4

,

5

,

10

,

11

,

15

The median requires the data set to be sorted from least to greatest.

2

Identify the middle value in the sorted list.

The middle value is

10

.

Since there is an odd number of values in the list (five days), the median is exactly the middle number.

Key Concept

Measures of Central Tendency (Median)

Question 33Question →

Determine the total gross earnings for the month based on the given compensation structure.

Fill in the blanks below

During the month of October, a medical equipment consultant recorded total sales of

\$45,000

. His compensation package includes a guaranteed monthly base salary of

\$1,200

. On top of this base, he earns a

5\%

commission on the first

\$20,000

of his sales, and an

8\%

commission on all sales volume beyond that initial

\$20,000

. Based on these figures, the consultant's total gross pay for October is $\$$.

Show answer & explanation

Answer

The consultant's total gross pay is $4,200.

The total gross pay is found by isolating the sales into two tiers: the first

20,000 earns a 5% commission (

1,000) and the remaining

25,000 earns an 8% commission (

2,000). Adding these two commissions to the

1,200 base salary yields a total gross pay of

4,200.

Step-by-Step Solution

1

Calculate the commission earned on the first tier of sales.

\$20,000 \times 0.05 = \$1,000

The consultant earns a 5% commission on the initial $20,000 of sales.

2

Determine the amount of sales that qualify for the second commission tier.

\$45,000 - \$20,000 = \$25,000

The higher commission rate only applies to the sales volume exceeding the first $20,000.

3

Calculate the commission earned on the second tier of sales.

\$25,000 \times 0.08 = \$2,000

The consultant earns an 8% commission on the remaining $25,000 of sales.

4

Sum the base salary and both commission amounts to find the total gross pay.

\$1,200 + \$1,000 + \$2,000 = \$4,200

Total earnings are the sum of the guaranteed salary and all commissions earned.

Key Concept

Calculating total earnings involving a base salary and a multi-tiered commission structure.

Question 34Question →

A cargo plane travels at a constant speed of $300\text{ miles per hour}$ . At this rate, how many miles does the plane travel in $20\text{ minutes}$ ?

15

60

100

6,000

Show answer & explanation

Answer: $100$

Answer

100

The correct answer correctly converts the travel time to hours by recognizing that

20\text{ minutes}

is exactly one-third (

\frac{1}{3}

) of an hour. Multiplying the rate of

300\text{ miles per hour}

by

\frac{1}{3}\text{ of an hour}

gives a total distance of

100\text{ miles}

.

Step-by-Step Solution

1

Convert the given time from minutes to hours.

20\text{ minutes} = \frac{20}{60}\text{ hours} = \frac{1}{3}\text{ hour}

The speed is given in miles per hour, so the time must be in hours before multiplying.

2

Calculate the distance using the formula Distance = Rate × Time.

300 \times \frac{1}{3} = 100\text{ miles}

Multiplying the constant rate by the elapsed time yields the total distance traveled.

Key Concept

Converting time units to match the rate units before calculating distance.

Estimated Time:45s

Question 35Question →

Match each technical passage describing military and aerospace engineering systems with the statement that expresses its explicitly stated main idea.

Click a left item, then click its matching right item

Show answer & explanation

Answer

Passage 1 matches with the statement on ultrasonic non-destructive testing identifying internal layer separations in aircraft structures; Passage 2 matches with the statement on frequency-hopping spread spectrum securing tactical radio communications against jamming; Passage 3 matches with the statement on electrolytic water-splitting generators serving as the primary oxygen source for submerged crews.

Each passage explicitly presents its central concept in an identifiable opening topic sentence. The first passage focuses on ultrasonic inspection as the premier method for uncovering hidden composite delamination; the second passage establishes frequency-hopping spread spectrum as the primary technology for securing tactical radio transmissions against jamming; and the third passage explicitly defines water electrolysis as the primary system supplying breathable oxygen to submerged submarine crews.

Step-by-Step Solution

1

Examine the first passage to locate its explicitly stated topic sentence.

The lead sentence states: 'Ultrasonic non-destructive inspection remains the most reliable method for detecting hidden subsurface delamination in aircraft composite structures.'

Explicit main ideas in technical passages are often located in the opening sentence, framing the entire discussion of acoustic wave reflections and integrity checks.

2

Examine the second passage to identify its central explicit assertion.

The lead sentence states: 'Tactical radio units maintain operational security in hostile electronic warfare environments through frequency-hopping spread spectrum technology.'

The supporting sentences elaborate on how frequency changes work and prevent jamming, reinforcing that the core claim is operational communication security via frequency hopping.

3

Examine the third passage to identify its explicit primary claim.

The lead sentence states: 'Submerged naval vessels rely primarily on water electrolysis units to maintain continuous breathable oxygen levels for the crew.'

Subsequent details describe electrochemical splitting and gas metering, which support the principal stated function of water electrolysis for oxygen generation.

4

Pair each technical passage with its corresponding paraphrased explicit main idea statement.

All three passages are paired accurately with their exact topic-level summaries.

Matching each explicit topic sentence ensures that central points are distinguished from specific supporting mechanisms.

Key Concept

Identifying explicitly stated main ideas in technical passages

Estimated Time:1m 30s

Question 36Question →

Read the three informational passages below regarding biological adaptations and engineering materials. Match each passage with the title that most accurately captures its central theme and scope.

Click a left item, then click its matching right item

Show answer & explanation

Answer

Passage 1 matches 'Counterillumination: Optical Concealment in Midwater Cephalopods', Passage 2 matches 'Pozzolanic Chemistry and the Longevity of Roman Marine Concrete', and Passage 3 matches 'Cavitation Shockwaves: The Snapping Shrimp's Acoustic Weapon'.

Each passage is paired with the title that directly reflects its primary subject and mechanism: Passage 1 focuses on cephalopod counterillumination camouflage, Passage 2 discusses the chemical durability of Roman pozzolanic concrete, and Passage 3 explains the generation of cavitation shockwaves by snapping shrimp.

Step-by-Step Solution

1

Analyze the central focus and supporting mechanism of Passage 1.

The text details how deep-sea squids use belly photophores to mimic sunlight and eliminate their shadows from predators below.

This establishes that the passage is about bioluminescent camouflage (counterillumination) in cephalopods.

2

Analyze the central focus and material mechanism of Passage 2.

The text focuses on Roman concrete made with pozzolanic volcanic ash that reacts with seawater to form strengthening crystals.

This identifies the topic as the chemical longevity of Roman marine concrete containing pozzolana.

3

Analyze the central focus and physical mechanism of Passage 3.

The text explains how snapping shrimp claws create cavitation bubbles that collapse into acoustic shockwaves to stun prey.

This confirms the topic as cavitation shockwaves used by snapping shrimp.

4

Match each analyzed passage to its corresponding specific title.

Pairing Passage 1 with the counterillumination title, Passage 2 with the pozzolanic concrete title, and Passage 3 with the cavitation shockwave title.

Each title uniquely and comprehensively encapsulates the main idea and scope of its corresponding passage without overgeneralizing.

Key Concept

Selecting the Best Title by identifying the core subject, primary mechanism, and scope of a passage.

Estimated Time:1m 30s

Question 37Question →

Match each fundamental principle of electrical current on the left with its correct physical definition or standard unit on the right.

Click a left item, then click its matching right item

Show answer & explanation

Answer

Electric Current pairs with the rate of electric charge flow; Ampere pairs with one coulomb per second; Electron Flow pairs with movement from negative to positive; Conventional Current pairs with flow from positive to negative.

Electric current is the rate at which charge passes a point in a circuit, measured in amperes (

1\text{ A} = 1\text{ C/s}

). In physical conductors, negatively charged electrons flow from the negative source terminal to the positive terminal (electron flow), whereas standard circuit analysis historically assumes flow from positive to negative (conventional current).

Step-by-Step Solution

1

Identify the definition of electric current and its unit of measurement

Electric current corresponds to the rate of charge flow through a conductor, and the ampere represents one coulomb of charge passing per second.

Current is mathematically

I = \frac{Q}{t}

, measured in amperes (

1\text{ A} = 1\text{ C/s}

).

2

Distinguish between the two directional models of current flow

Electron flow moves from negative to positive (the actual movement of electrons in conductors), whereas conventional current is modeled from positive to negative.

Electrons carry negative charge and are repelled by negative potential, whereas conventional current was standardized before the discovery of the electron.

Key Concept

Electric current is the time rate of charge flow, measured in amperes (

1\text{ A} = 1\text{ C/s}

), with electron flow occurring from negative to positive and conventional current modeled from positive to negative.

Estimated Time:45s

Question 38Question →

Complete the statement regarding the electrical parameters of the DC circuit by calculating the missing resistance and current values.

Fill in the blanks below

An indicator lamp with a constant resistance draws

150\text{ mA}

when connected to a

24\text{ V}

direct current source. The internal resistance of the lamp is

\Omega

, and if the supply voltage is increased to

36\text{ V}

, the current drawn by the lamp will increase to

\text{mA}

.

Show answer & explanation

Answer

The internal resistance of the lamp is 160 Ω, and the new current at 36 V is 225 mA.

Converting

150\text{ mA}

to standard amperes yields

0.15\text{ A}

. Using Ohm's law (

R = V / I

), the internal resistance is

24\text{ V} / 0.15\text{ A} = 160\ \Omega

. When the voltage rises to

36\text{ V}

, applying

I = V / R

produces

36\text{ V} / 160\ \Omega = 0.225\text{ A}

, which corresponds to

225\text{ mA}

.

Step-by-Step Solution

1

Convert the nominal current from milliamperes to base amperes

I_1 = 150\text{ mA} = 0.15\text{ A}

Ohm's law requires base SI units (amperes, volts, ohms) for direct calculation.

2

Calculate the constant internal resistance using Ohm's law

R = \frac{V_1}{I_1} = \frac{24\text{ V}}{0.15\text{ A}} = 160\ \Omega

Resistance equals voltage divided by current.

3

Calculate the new current at the elevated voltage level

I_2 = \frac{V_2}{R} = \frac{36\text{ V}}{160\ \Omega} = 0.225\text{ A} = 225\text{ mA}

Current across a fixed resistance is directly proportional to applied voltage (

I = V / R

).

Key Concept

Ohm's Law (

V = I \times R

) and Direct Proportionality in DC Circuits

Estimated Time:45s

Question 39Question →

Match each capacitor specification or parameter with its correct electrical description.

Click a left item, then click its matching right item

Show answer & explanation

Answer

Dielectric matches the insulating material between conductive plates; Working Voltage (WVDC) matches the maximum continuous direct-current potential difference before breakdown; Capacitive Reactance matches the frequency-dependent opposition to alternating current; and Farad matches the base unit of electrostatic capacity storing one coulomb per volt.

Each capacitor term corresponds to its fundamental physical or operational definition: the dielectric is the insulating layer, working voltage is the maximum safe direct voltage before breakdown, capacitive reactance is the frequency-dependent opposition to alternating current, and the farad is the SI unit representing charge storage per volt.

Step-by-Step Solution

1

Identify the physical insulating component inside a capacitor.

The dielectric is the insulating material placed between conductive plates that establishes permittivity and prevents direct conduction.

Capacitors store energy in an electrostatic field across an insulator, which is termed the dielectric.

2

Identify the maximum safe operating voltage parameter.

Working Voltage (WVDC) indicates the maximum continuous direct voltage the component can withstand without dielectric puncture.

Exceeding this rating causes dielectric breakdown, arcing, and component destruction.

3

Identify the parameter defining AC opposition.

Capacitive Reactance (

X_C

) is the opposition offered to alternating current, calculated as

X_C = \frac{1}{2\pi f C}

.

Capacitors impede alternating current depending inversely on the signal frequency and capacitance value.

4

Identify the primary electrical unit of capacitance.

The Farad (

F

) is the standard unit of capacitance, where

1\text{ F} = 1\text{ C/V}

.

Capacitance (

C

) equals stored electrostatic charge (

Q

) divided by the voltage difference (

V

).

Key Concept

Capacitor Construction, Ratings, Units, and AC Behavior

Estimated Time:1m 0s

Question 40Question →

An industrial machine uses a safety interlock limit switch on an access guard door. When the guard door is fully closed, it pushes down a spring-loaded plunger to complete the control circuit. When the door is opened, the plunger returns to its unactuated resting position, breaking the circuit and cutting power. Which switch configuration describes this device?

A normally open (NO) momentary switch that completes the circuit only when mechanically actuated

A normally closed (NC) momentary switch that maintains electrical continuity in its resting position

A double-pole, double-throw (DPDT) toggle switch configured for polarity reversal

A thermal bimetallic breaker switch that opens automatically when current exceeds a set limit

Show answer & explanation

Answer: A normally open (NO) momentary switch that completes the circuit only when mechanically actuated

Answer

A normally open (NO) momentary switch that completes the circuit only when mechanically actuated

The 'normal' state of any electrical switch or relay contact is defined as its resting condition when no external mechanical force or coil energization is applied. In this access guard configuration, opening the door releases the plunger to its resting position, opening the circuit (infinite resistance, stopping current flow). Closing the door applies the mechanical force necessary to hold the contacts closed. Therefore, the device is a normally open (NO) momentary switch.

Step-by-Step Solution

1

Identify the unactuated (resting) electrical state of the switch contacts.

When the guard door opens, the plunger is released back to its resting state, opening the electrical circuit.

By industry definition, the designation of a switch (Normally Open vs. Normally Closed) is determined by its state when no external force or actuation is present.

2

Determine the switch classification corresponding to this resting state and action.

Because the switch is open in its resting position and only conducts when held down by the door, it is a Normally Open (NO) momentary switch.

Momentary NO contacts require continuous physical force to maintain continuity and spring back to an open circuit when that force is removed.

Key Concept

Switch Contact Normal States (NO vs. NC) and Momentary Interlock Principles

Estimated Time:45s

Answer

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