All practice questions

Question 81Question →

A technician is designing a solid metal firewall to separate a hot engine compartment from a passenger cabin. Which of the following modifications will specifically reduce the rate of heat transferred by conduction through the solid wall?

Increasing the overall thickness of the solid metal wall.

Using a metal alloy that has a higher thermal conductivity.

Polishing the engine-facing surface to a shiny, reflective finish.

Cutting open vents into the wall to allow air circulation.

Show answer & explanation

Answer: Increasing the overall thickness of the solid metal wall.

Answer

Increasing the overall thickness of the solid metal wall.

Heat conduction through a solid occurs as energetic molecules transfer kinetic energy to neighboring molecules. By increasing the thickness of the wall, the heat has to travel a greater physical distance. This decreases the temperature gradient across the material, which proportionally slows down the rate of conductive heat transfer.

Step-by-Step Solution

1

Identify the mode of heat transfer being tested.

The question specifically asks about conduction, which is the transfer of heat through direct contact within a solid material.

Different modifications affect different modes of heat transfer (conduction, convection, and radiation).

2

Evaluate how physical changes affect conduction.

Conduction rate is reduced by increasing the distance the heat must travel (thickness) or by using a material with lower thermal conductivity.

The rate of thermal conduction is inversely proportional to the thickness of the material.

3

Analyze the given options against conduction principles.

Increasing thickness reduces conduction. Reflective surfaces affect radiation. Vents introduce convection. Higher conductivity increases conduction.

This confirms that only increasing the thickness specifically addresses and reduces conduction.

Key Concept

Factors Affecting Thermal Conduction

Question 82Question →

Based on Newton's First Law of motion, how does an unbelted passenger's body tend to react relative to a vehicle in various situations? Pair each specific vehicle maneuver with the passenger's corresponding initial directional tendency.

Click a left item, then click its matching right item

Show answer & explanation

Answer

Braking pairs with sliding forward; accelerating pairs with pressing backward; turning left pairs with leaning right; and steady cruising pairs with no directional change.

Each pair correctly applies the concept of inertia. A passenger's body will always resist changes in the vehicle's speed or direction, maintaining its current state of motion or rest until an unbalanced force acts upon it.

Step-by-Step Solution

1

Analyze the sudden braking scenario.

The passenger's body tends to slide or lean forward.

Due to inertia, an object in motion tends to stay in motion. The car slows, but the passenger's body wants to continue at its original speed.

2

Analyze the rapid acceleration scenario.

The passenger's body is pressed backward into the seat.

An object at rest tends to stay at rest. As the car shoots forward, the passenger's mass resists the forward motion, causing them to lag behind.

3

Analyze the sharp left turn scenario.

The passenger's body tends to slide or lean to the right.

Objects in motion tend to move in a straight line. The car curves left, but the body attempts to travel straight, appearing to move right relative to the vehicle's interior.

4

Analyze the steady cruising scenario.

The passenger's body experiences no change in perceived direction.

With a constant velocity and no change in direction, the net force is zero, and the body remains in equilibrium with the car.

Key Concept

Newton's First Law (Inertia)

Question 83Question →

While welding a steel bracket, a fabricator notices that the heat travels along the length of the solid metal, eventually making the opposite unheated end too hot to handle. Which mechanism is primarily responsible for transferring heat through this solid metal component?

Conduction

Convection

Radiation

Thermal expansion

Show answer & explanation

Answer: Conduction

Answer

Conduction is the correct mechanism for heat transfer through a solid.

Conduction is the physical process where heat energy is transferred through a solid material by direct molecular collision. When the torch heats one end of the steel bracket, the energized molecules vibrate and pass the heat energy to adjacent molecules all the way down the metal.

Step-by-Step Solution

1

Analyze the physical state of the medium transferring the heat in the scenario.

The heat is moving through a 'solid metal component' (the steel bracket).

Different modes of heat transfer operate in different mediums (solids, fluids, or vacuums).

2

Identify the heat transfer method associated with solid materials.

Heat transfer within a solid object due to a temperature gradient occurs via direct molecular interaction, which is defined as conduction.

Convection requires a circulating fluid, and radiation does not rely on a solid medium, leaving conduction as the correct process.

Key Concept

Heat Transfer via Conduction

Question 84Question →

Four identical airtight containers, each with a total volume of $1 \text{ m}^3$ , are placed in different situations. Arrange these situations in order from the GREATEST buoyant force acting on the container to the LEAST buoyant force.

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct order from greatest to least buoyant force is: fully submerged in saltwater, fully submerged in freshwater, half-submerged in freshwater, and sitting on dry land.

Buoyant force is determined by the weight of the displaced fluid. A fully submerged object in saltwater displaces the heaviest amount of fluid. A fully submerged object in freshwater displaces slightly less weight because freshwater is less dense. A half-submerged object displaces only half the volume of water, yielding an even smaller force. An object on dry land displaces no water, experiencing zero buoyant force.

Step-by-Step Solution

1

Recall Archimedes' Principle.

The buoyant force is equal to the weight of the displaced fluid (

F_b = \text{fluid density} \times \text{volume displaced} \times \text{gravity}

).

This formula identifies the two main variables that affect buoyant force: the density of the fluid and the volume of fluid displaced.

2

Compare the volume of fluid displaced in each scenario.

Both completely submerged scenarios displace the full

1 \text{ m}^3

volume. The half-submerged floating scenario displaces

0.5 \text{ m}^3

. The dry land scenario displaces

0 \text{ m}^3

.

Volume displaced is the primary factor in determining the magnitude of buoyant force.

3

Compare the fluid densities for the fully submerged scenarios.

Saltwater has a higher density than standard freshwater. Therefore,

1 \text{ m}^3

of saltwater weighs more than

1 \text{ m}^3

of freshwater.

When displaced volumes are equal, the fluid with the higher density exerts a greater buoyant force.

4

Order the scenarios based on the weight of the displaced fluid.

Submerged in Saltwater > Submerged in Freshwater > Half-submerged in Freshwater > Dry land.

This sequence correctly ranks the situations from greatest to least buoyant force.

Key Concept

According to Archimedes' Principle, buoyant force is directly proportional to both the volume of fluid displaced and the density of that fluid.

Question 85Question →

An engineer simultaneously launches a $1\text{ kg}$ testing sphere and a $5\text{ kg}$ testing sphere vertically upward with the exact same initial velocity. At the precise instant they reach their maximum heights (apexes), which of the following statements accurately describes their vertical accelerations? (Assume $g = 9.8\text{ m/s}^2$ and ignore air resistance).

The

5\text{ kg}

sphere experiences a greater downward acceleration because its heavier weight exerts a stronger pull.

Both spheres experience a downward acceleration of

0\text{ m/s}^2

because they are momentarily stationary.

The

1\text{ kg}

sphere experiences a greater downward acceleration because its smaller mass provides less inertia against gravity.

Both spheres experience a constant downward acceleration of

9.8\text{ m/s}^2

.

Show answer & explanation

Answer: Both spheres experience a constant downward acceleration of $9.8\text{ m/s}^2$ .

Answer

Both spheres experience a constant downward acceleration of

9.8\text{ m/s}^2

.

In the absence of air resistance, the acceleration due to gravity (

g

) is constant for all objects, regardless of their mass. When an object reaches its highest point, its velocity is momentarily zero, but the gravitational force continues to pull on it without interruption, causing a constant downward acceleration of

9.8\text{ m/s}^2

.

Step-by-Step Solution

1

Analyze the forces acting on the spheres at the apex.

Gravity is the only force acting on both spheres, meaning they are in a continuous state of free fall.

Because air resistance is ignored, gravitational pull is the sole factor determining acceleration throughout the entire flight.

2

Determine the effect of mass on gravitational acceleration.

The acceleration due to gravity is independent of an object's mass.

Although a heavier object experiences a greater gravitational force (weight), it also has proportionally more inertia (mass). These factors cancel out (

a = F/m

), resulting in the same constant acceleration of

9.8\text{ m/s}^2

for both spheres.

3

Evaluate the effect of momentary zero velocity on acceleration.

Acceleration remains

9.8\text{ m/s}^2

downward even when vertical velocity is momentarily

0\text{ m/s}

.

Acceleration is the rate of change of velocity. Gravity continuously changes the upward velocity to a downward velocity, passing through zero without the acceleration itself ever dropping to zero.

Key Concept

Constant Gravitational Acceleration in Free Fall

Question 86Question →

A metal fabricator is designing a custom hand tool to manipulate red-hot steel parts in a forge. To ensure the handle stays safe to touch, the designer must minimize the rate of conductive heat transfer from the hot jaws along the tool's solid shaft. Which of the following shaft designs will most effectively restrict this heat conduction?

A long shaft with a reduced cross-sectional area, made from a material with low thermal conductivity.

A thick, massive shaft designed to absorb the thermal energy before it reaches the handle.

A short, wide shaft designed to rapidly dissipate the heat into the surrounding air.

A standard shaft coated with a highly reflective metallic finish to bounce the heat back toward the jaws.

Show answer & explanation

Answer: A long shaft with a reduced cross-sectional area, made from a material with low thermal conductivity.

Answer

The most effective design is a long shaft with a reduced cross-sectional area, made from a material with low thermal conductivity.

The rate of heat conduction through a solid object is directly proportional to its cross-sectional area and inversely proportional to its length. Therefore, a longer, thinner shaft made of an insulating material provides the greatest physical resistance to conductive heat flow.

Step-by-Step Solution

1

Identify the primary mechanism of heat transfer being restricted.

The problem specifies minimizing 'conductive heat transfer' through a solid shaft.

Different modes of heat transfer (conduction, convection, radiation) require different mitigation strategies.

2

Recall the factors that affect the rate of thermal conduction in a solid.

Conduction rate depends on the material's thermal conductivity, the cross-sectional area (thickness), and the length of the pathway.

Fourier's law of heat conduction dictates how physical dimensions influence heat flow.

3

Determine how to modify these factors to minimize conduction.

To restrict heat flow, the length must be increased (longer path), the cross-sectional area must be decreased (narrower path), and a material with low conductivity must be used.

Heat conduction is directly proportional to area and inversely proportional to length.

Key Concept

Factors Affecting Thermal Conduction

Question 87Question →

A mechanic swings a $2 \text{ lb}$ hammer, striking a $100 \text{ lb}$ stationary steel anvil. During the brief fraction of a second of contact, the hammer rapidly decelerates to a stop while the anvil remains practically motionless. Which of the following statements correctly compares the interaction forces between the hammer and the anvil during the exact moment of impact?

The force the hammer exerts on the anvil is exactly equal in magnitude to the force the anvil exerts on the hammer.

The anvil exerts a greater force on the hammer because the hammer undergoes a much larger change in acceleration during the impact.

The hammer exerts a greater force on the anvil because the heavy mass and inertia of the anvil prevent it from generating an equal reaction force.

The forces would only be equal if both objects had the same mass; otherwise, the object with greater mass exerts the dominant force.

Show answer & explanation

Answer: The force the hammer exerts on the anvil is exactly equal in magnitude to the force the anvil exerts on the hammer.

Answer

The force the hammer exerts on the anvil is exactly equal in magnitude to the force the anvil exerts on the hammer.

According to Newton's Third Law of Motion, forces always occur in equal and opposite pairs. When the hammer strikes the anvil, the interaction consists of the hammer pushing on the anvil and the anvil pushing on the hammer. These two forces are simultaneously created and are always perfectly equal in magnitude, regardless of the fact that the anvil is

50

times heavier than the hammer or that the hammer is rapidly decelerating. The difference in their observed movement is due to their differing masses (Newton's Second Law), not a difference in the force applied.

Step-by-Step Solution

1

Identify the two interacting bodies in the mechanical system.

The interacting bodies are the

2 \text{ lb}

hammer and the

100 \text{ lb}

anvil.

Newton's Third Law always applies to the mutual interaction between exactly two distinct objects.

2

Apply Newton's Third Law to the specific moment of impact.

The force of the hammer on the anvil (

F_{H \to A}

) must equal the negative force of the anvil on the hammer (

-F_{A \to H}

).

For every action force, there is an equal and opposite reaction force, regardless of mass or velocity.

3

Differentiate between the interaction force and the resulting acceleration.

Both objects experience the exact same magnitude of force, but the hammer decelerates rapidly because it has a much smaller mass.

Newton's Second Law (

F = ma

) explains the differing accelerations, while Newton's Third Law dictates the equal forces.

Key Concept

Action and reaction forces between two interacting bodies are always perfectly equal in magnitude and opposite in direction, entirely independent of the objects' masses or rates of acceleration.

Estimated Time:1m 30s

Question 88Question →

Arrange the following steps in the correct chronological sequence for shrink-fitting a metal collar onto a steel shaft.

Drag items to arrange them in the correct order

Show answer & explanation

Answer

First heat the collar, then slide it over the shaft, allow the assembly to cool, and finally the collar contracts to create a tight grip.

In a shrink-fitting process, the outer component (the collar) must be heated first so that thermal expansion increases its internal diameter. While it is expanded, it is slid over the inner component (the shaft). As the assembly cools to room temperature, the outer component undergoes thermal contraction, shrinking tightly around the shaft to create a secure mechanical grip.

Step-by-Step Solution

1

Identify the first step required to make the parts fit together.

Heat the metal collar to increase its internal diameter.

Heating causes thermal expansion, creating the necessary clearance for the shaft.

2

Identify the physical assembly step.

Slide the heated collar over the shaft.

The parts must be assembled quickly while the collar is still expanded.

3

Identify the next phase of the physical process.

Allow the assembled collar and shaft to cool.

Removing the heat source allows the metal to return to its original temperature.

4

Identify the final mechanical outcome.

The collar contracts and grips the shaft tightly.

Thermal contraction shrinks the collar's diameter, creating a strong interference fit.

Key Concept

Thermal Expansion and Contraction

Question 89Question →

Arrange the following materials in order from the BEST thermal insulator (slowest rate of heat transfer) to the WORST thermal insulator (fastest rate of heat transfer).

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct order from best to worst thermal insulator is: Fiberglass insulation, Solid wood, Window glass, Copper piping.

Fiberglass traps air, giving it the highest insulation value. Wood is naturally porous and insulates moderately well. Glass is a poor insulator but still restricts heat more than metals. Copper is highly conductive, making it the worst insulator.

Step-by-Step Solution

1

Identify the material known for intentionally blocking heat transfer.

Fiberglass insulation is designed specifically to prevent heat loss or gain in buildings, placing it first.

Trapped air within the fiberglass matrix provides exceptional resistance to thermal conduction.

2

Evaluate the remaining non-metal materials.

Solid wood is placed second, and window glass is placed third.

Wood's internal cellular structure acts as a moderate insulator (often used for pot handles), whereas solid glass is dense and transfers heat more readily.

3

Identify the material known for thermal conductivity.

Copper piping is placed last.

Metals like copper have highly mobile electrons that transfer kinetic energy (heat) rapidly, making them excellent conductors and terrible insulators.

Key Concept

Thermal conductivity dictates how quickly heat moves through a material. Good insulators (like air-trapping fiberglass) have low conductivity, while conductors (like metals) have high conductivity.

Question 90Question →

Based on Newton's First Law of Motion (Inertia), how does each object behave in its respective environment? Match each physical scenario on the left with the correct description of its resulting motion or behavior on the right.

Click a left item, then click its matching right item

Show answer & explanation

Answer

The satellite matches with maintaining a constant trajectory (no net force); the astronaut matches with shifting right relative to the cabin (inertia of direction); the water droplet matches with flying off in a straight tangent line (removal of centripetal force); and the granite block matches with exhibiting extreme resistance to motion (high mass means high inertia).

Each correct pairing directly applies Newton's First Law (Inertia) to distinct physical environments. Objects free of external forces (the satellite) travel at constant velocity. Objects in accelerating reference frames (the turning spacecraft) appear to move opposite the acceleration because they maintain their straight-line inertia. Objects released from circular paths (the droplet) travel straight tangentially. Finally, massive objects (the granite block) inherently resist acceleration due to their large mass, which is completely independent of surface friction.

Step-by-Step Solution

1

Analyze the deep-space satellite scenario.

It experiences zero net external forces (no gravity, no drag).

According to the First Law, an object with no net force acting upon it will maintain a constant velocity (speed and direction) indefinitely.

2

Analyze the astronaut in the turning spacecraft.

The spacecraft accelerates to the left, but no lateral force acts directly on the unbelted astronaut.

The astronaut's inertia keeps them moving in their original straight line, making them appear to 'fall' toward the right wall of the turning ship.

3

Analyze the detaching water droplet.

The physical connection providing inward (centripetal) force is broken.

Without a net force pulling it into a curve, the droplet obeys inertia and continues in a straight line tangent to its path.

4

Analyze the stationary granite block on a frictionless surface.

Friction is zero, but the object's mass is extremely high.

Mass is the physical measure of inertia; greater mass means greater intrinsic resistance to acceleration, regardless of friction.

Key Concept

Inertia is the inherent tendency of an object to resist changes to its state of motion or rest. This resistance is directly proportional to the object's mass and dictates that an object will maintain a constant velocity unless acted upon by a net external force.

Question 91Question →

A civil engineer needs to earn exactly $\$840$ in simple interest over a $10$ -month period to offset the cost of new drafting software. If a local credit union offers a business savings account with a $4.8\%$ annual simple interest rate, how much principal must be deposited into the account?

\$1,750

\$2,100

\$17,500

\$21,000

Show answer & explanation

Answer: $\$21,000$

Answer

The correct principal amount to deposit is

\$21,000

.

The correct answer accurately uses the simple interest formula by first converting the 10-month period into a fraction of a year (

10/12

or

5/6

). Multiplying the annual rate of

0.048

by

5/6

yields

0.04

. Dividing the target interest of

\$840

by

0.04

correctly determines the required principal.

Step-by-Step Solution

1

Identify the known variables from the problem.

Interest (

I

) =

\$840

, annual rate (

r

) =

4.8\% = 0.048

, and time (

t

) =

10

months.

Organizing the given information clarifies what is needed for the simple interest formula.

2

Convert the time period from months to years because the interest rate is an annual rate.

t = \frac{10}{12} = \frac{5}{6} \text{ years}

Standard simple interest formulas require time to be in the same unit as the rate.

3

Substitute the values into the simple interest formula

I = P \times r \times t

to set up the equation.

840 = P \times 0.048 \times \frac{5}{6}

This establishes the mathematical relationship to isolate the unknown principal amount (

P

).

4

Simplify the equation and solve for the principal.

840 = P \times 0.04 \Rightarrow P = \frac{840}{0.04} = 21000

Dividing both sides by the calculated coefficient isolates

P

and yields the final answer.

Key Concept

Solving for Principal using the Simple Interest Formula

Question 92Question →

An independent contractor installs smart home systems. He earns a base rate of $\$18.50$ per hour for a standard $40$ -hour workweek. For any overtime hours worked beyond $40$ hours, he earns time-and-a-half. He also receives a $5\%$ commission on all equipment sales he makes. If he worked $48$ hours this week and sold $\$3,200$ worth of equipment, what are his total gross earnings for the week?

\$382.00

\$1,122.00

\$2,562.00

\$4,322.00

Show answer & explanation

Answer: \$1,122.00

Answer

The total gross earnings for the week are

\$1,122.00

.

The correct answer accurately accounts for all three payment streams. The base pay for 40 hours is

40 \times \$18.50 = \$740.00

. The overtime pay for the 8 extra hours is calculated at time-and-a-half (

18.50 \times 1.5 = \$27.75/hr

), which yields

8 \times \$27.75 = \$222.00

. The 5% commission on

\$3,200

in sales is

0.05 \times 3,200 = \$160.00

. Adding these together (

\$740 + \$222 + \$160

) results in

\$1,122.00

.

Step-by-Step Solution

1

Calculate the standard base pay for

40

hours.

40 \times \$18.50 = \$740.00

The contractor earns a fixed hourly rate for the first 40 hours worked.

2

Calculate the number of overtime hours and the overtime rate.

Overtime hours:

48 - 40 = 8

. Overtime rate:

\$18.50 \times 1.5 = \$27.75

.

Overtime is paid at 'time-and-a-half' for any hours worked beyond the standard 40 hours.

3

Calculate the total overtime pay.

8 \times \$27.75 = \$222.00

Multiply the overtime hours by the higher overtime pay rate.

4

Calculate the commission earned on equipment sales.

0.05 \times \$3,200 = \$160.00

The commission is 5% (or 0.05) of the total sales volume.

5

Sum all three components to find the total gross earnings.

\$740.00 + \$222.00 + \$160.00 = \$1,122.00

Gross earnings are the sum of base pay, overtime pay, and commission before any deductions.

Key Concept

Calculating Gross Pay with Overtime and Commission

Question 93Question →

Calculate the missing financial values for the equipment purchase described below.

Fill in the blanks below

A veterinary clinic is purchasing a new digital ultrasound machine originally listed at

\$6,500

. The medical supplier is offering a

20\%

promotional discount. Because the clinic agrees to pick up the equipment themselves, they receive an additional

5\%

discount off the already reduced price.

The total discount amount the clinic receives is $\$

and the final price paid for the machine is

\$$.

Show answer & explanation

Answer

The total discount amount is

1,560 and the final price is

4,940.

The first discount of 20% saves

1,300, bringing the price down to

5,200. The second discount of 5% applies to this new

5,200 balance, saving an additional

260. Subtracting the second discount from

5,200 yields a final price of

4,940. The total amount saved combines both discounts:

1,300 +

260 = $1,560.

Step-by-Step Solution

1

Calculate the amount of the first discount.

$1,300

Multiply the original price ($6,500) by the first discount rate (20% or 0.20).

2

Calculate the price after the first discount.

$5,200

Subtract the first discount (

1,300) from the original price (

6,500).

3

Calculate the amount of the second discount.

$260

Multiply the reduced price ($5,200) by the additional discount rate (5% or 0.05).

4

Determine the final price.

$4,940

Subtract the second discount (

260) from the previously reduced price (

5,200).

5

Determine the total discount amount.

$1,560

Add the first discount (

1,300) and the second discount (

260).

Key Concept

Successive Discounts

Question 94Question →

Simplify the given algebraic expression by factoring it completely, then determine the missing positive integer values.

Fill in the blanks below

The completely factored form of the expression

2y^2 - 10y - 28

can be written in the format

2(y - a)(y + b)

, where

a

and

b

are positive integers.

The value of

a

is and the value of

b

is .

Show answer & explanation

Answer

The value of a is 7 and the value of b is 2.

Factoring out the GCF of 2 yields

2(y^2 - 5y - 14)

. The trinomial

y^2 - 5y - 14

factors into

(y - 7)(y + 2)

. Matching this with the given structure

2(y - a)(y + b)

shows that the positive integers are

a = 7

and

b = 2

.

Step-by-Step Solution

1

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

2(y^2 - 5y - 14)

Identifying and factoring out the GCF of 2 simplifies the polynomial, revealing a trinomial with a leading coefficient of 1, which is easier to factor.

2

Factor the resulting quadratic trinomial

y^2 - 5y - 14

.

The binomial factors are

(y - 7)

and

(y + 2)

.

To factor

y^2 - 5y - 14

, we need to find two numbers that multiply to the constant term (-14) and add up to the middle term's coefficient (-5). The numbers -7 and 2 satisfy both conditions (-7 * 2 = -14 and -7 + 2 = -5).

3

Write the complete factorization and match it to the provided template

2(y - a)(y + b)

.

The expression is

2(y - 7)(y + 2)

. Matching this to the template gives

a = 7

and

b = 2

.

Comparing the forms directly, the term subtracted from y is 7 (so a = 7) and the term added to y is 2 (so b = 2).

Key Concept

Factoring polynomials using the Greatest Common Factor (GCF) and factoring quadratic trinomials.

Question 95Question →

An armory quartermaster is ordering gun solvent for weapons maintenance. The solvent is available in four different packaging options. Arrange the options from the lowest price per gallon to the highest price per gallon.

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct order from lowest to highest price per gallon is: 2-quart bottle, 5-gallon bucket, 4-gallon case, 1-pint bottle.

By converting all package volumes into gallons, we can compare their unit prices accurately. The

2

-quart bottle costs

\$12.00

per gallon, the

5

-gallon bucket costs

\$13.00

per gallon, the

4

-gallon case costs

\$13.50

per gallon, and the

1

-pint bottle costs

\$14.00

per gallon. Arranging them from lowest to highest yields this sequence.

Step-by-Step Solution

1

Calculate the unit price for the

2

-quart bottle.

\$12.00

per gallon

There are

4

quarts in a gallon, making

2

quarts equivalent to

0.5

gallons.

\$6.00 \div 0.5 = \$12.00

per gallon.

2

Calculate the unit price for the

5

-gallon bucket.

\$13.00

per gallon

Dividing the total cost by the number of gallons:

\$65.00 \div 5 = \$13.00

per gallon.

3

Calculate the unit price for the

4

-gallon case.

\$13.50

per gallon

Dividing the total cost by the number of gallons:

\$54.00 \div 4 = \$13.50

per gallon.

4

Calculate the unit price for the

1

-pint bottle.

\$14.00

per gallon

There are

8

pints in a gallon. Multiplying the pint price by

8

gives the gallon price:

\$1.75 \times 8 = \$14.00

per gallon.

5

Order the unit prices from lowest to highest.

\$12.00

,

\$13.00

,

\$13.50

,

\$14.00

The question asks for the options to be arranged in ascending order based on their price per gallon.

Key Concept

Converting various liquid volume measurements to a common unit (gallons) in order to accurately compare unit prices.

Question 96Question →

Determine the exact ordered pair $(x, y)$ that satisfies the system of linear equations shown below.

4x - 3y = 25

x + 2y = 9

Enter your answer in the format $(x, y)$ .

Show answer & explanation

Answer: (7, 1); (7,1); 7, 1; 7,1

Answer

The correct ordered pair is (7, 1).

The correct solution is (7, 1). To verify, substitute x = 7 and y = 1 into both original equations. For the first equation: 4(7) - 3(1) = 28 - 3 = 25. For the second equation: 7 + 2(1) = 7 + 2 = 9. Since both equations hold true simultaneously, (7, 1) is the accurate solution.

Step-by-Step Solution

1

Isolate the variable

x

in the second equation.

x = 9 - 2y

The second equation has an

x

with a coefficient of 1, making it the most straightforward variable to isolate for substitution without creating fractions.

2

Substitute the expression for

x

into the first equation.

4(9 - 2y) - 3y = 25

This creates a single equation containing only the variable

y

, which can be solved algebraically.

3

Distribute the 4 and combine like terms.

36 - 8y - 3y = 25 \Rightarrow 36 - 11y = 25

Simplifying the equation prepares it for isolating the

y

term.

4

Solve for

y

by subtracting 36 from both sides and dividing by -11.

-11y = -11 \Rightarrow y = 1

This determines the numerical value for the y-coordinate of the solution.

5

Substitute

y = 1

back into the isolated

x

equation from Step 1.

x = 9 - 2(1) \Rightarrow x = 9 - 2 = 7

This provides the corresponding x-coordinate to complete the ordered pair.

Key Concept

Solving Systems of Linear Equations by Substitution

Question 97Question →

A circular swimming pool has a circumference of $20\pi$ feet. What is the diameter of the pool in feet?

Show answer & explanation

Answer: 20

Answer

20

The formula for the circumference of a circle is

C = \pi d

. By substituting the given circumference of

20\pi

into the formula, we get

20\pi = \pi d

. Dividing both sides by

\pi

yields a diameter of

20

feet.

Step-by-Step Solution

1

Identify the formula for the circumference of a circle.

C = \pi d

, where

C

is circumference and

d

is diameter.

This formula directly relates the given circumference to the unknown diameter.

2

Substitute the given value into the formula.

20\pi = \pi d

The problem states the circumference is

20\pi

feet.

3

Solve for the diameter (

d

).

d = 20

Dividing both sides of the equation by

\pi

isolates the value of

d

.

Key Concept

Circumference of a circle

Estimated Time:45s

Question 98Question →

Determine the missing side length based on the geometric properties of a right triangle.

Fill in the blanks below

In a right triangle, the two shorter sides (legs) measure

9

cm and

12

cm. According to the Pythagorean theorem, the length of the hypotenuse is cm.

Show answer & explanation

Answer

15

By applying the Pythagorean theorem (

a^2 + b^2 = c^2

) to a right triangle with legs of

9

and

12

, we square both numbers to get

81

and

144

. Their sum is

225

. The hypotenuse is the square root of

225

, which is

15

.

Step-by-Step Solution

1

Identify the known sides of the right triangle.

The legs (let's call them

a

and

b

) measure

9

cm and

12

cm.

The Pythagorean theorem requires knowing the lengths of two sides to find the third.

2

Set up the Pythagorean theorem equation.

a^2 + b^2 = c^2

, which becomes

9^2 + 12^2 = c^2

.

This formula relates the squares of the legs to the square of the hypotenuse (

c

).

3

Square the lengths of the two legs and add them together.

81 + 144 = 225

This calculation determines the value of

c^2

.

4

Take the square root of the sum to find the hypotenuse.

\sqrt{225} = 15

Since

c^2 = 225

, taking the square root gives the actual length of the hypotenuse (

c

).

Key Concept

Pythagorean Theorem

Question 99Question →

A rectangular swimming pool measures $40$ feet in length and $30$ feet in width. A concrete deck of uniform width is built around the entire outer edge of the pool. If the area of the concrete deck alone is $296$ square feet, what is the width of the deck, in feet?

Show answer & explanation

Answer: 2

Answer

2

The correct width is found by setting up an equation for the total area. The deck adds its width to both sides of the pool's length and width, making the overall dimensions

40 + 2w

and

30 + 2w

. The area of the deck alone is the total area minus the pool's area. Solving the algebraic equation

(40 + 2w)(30 + 2w) - 1200 = 296

yields a positive root of

w = 2

.

Step-by-Step Solution

1

Calculate the area of the pool.

40 \times 30 = 1200

square feet.

You need the baseline area of the inner rectangle before the deck is added.

2

Write algebraic expressions for the outer dimensions of the pool and deck combined.

Length

= 40 + 2w

, Width

= 30 + 2w

.

The deck has a uniform width

w

, which adds to both the top/bottom and left/right sides of the pool.

3

Formulate an equation for the area of the deck.

(40 + 2w)(30 + 2w) - 1200 = 296

.

The area of the deck alone is equal to the total area (pool plus deck) minus the area of the pool.

4

Expand the binomials and simplify into a standard quadratic equation.

1200 + 140w + 4w^2 - 1200 = 296 \implies 4w^2 + 140w - 296 = 0

.

Standard quadratic form (

ax^2 + bx + c = 0

) is required to solve for the unknown variable

w

.

5

Solve the quadratic equation for

w

.

Divide by

4

to get

w^2 + 35w - 74 = 0

. Factoring yields

(w + 37)(w - 2) = 0

, so

w = 2

.

Finding the positive root gives the actual physical width of the concrete deck.

Key Concept

Calculating the area of composite rectangular figures and solving quadratic equations derived from expanding geometric boundaries.

Question 100Question →

When measuring the operating potential difference across an active $120\text{ V}$ AC ventilation fan motor, which multimeter dial setting and lead connection configuration should be used?

Connect the meter in series with the motor, insert the red lead into the 10A jack, and set the dial to AC current (

A\sim

).

Connect the meter in parallel across the motor terminals, insert the red lead into the

\text{V}/\Omega

jack, and set the dial to AC voltage (

\text{V}\sim

).

Connect the meter in parallel across the motor terminals, insert the red lead into the

\text{V}/\Omega

jack, and set the dial to DC voltage (

\text{V}\overline{--}

).

Connect the meter in series with the motor, insert the red lead into the

\text{V}/\Omega

jack, and set the dial to AC voltage (

\text{V}\sim

).

Show answer & explanation

Answer: Connect the meter in parallel across the motor terminals, insert the red lead into the $\text{V}/\Omega$ jack, and set the dial to AC voltage ( $\text{V}\sim$ ).

Answer

Connect the meter in parallel across the motor terminals, insert the red lead into the

\text{V}/\Omega

jack, and set the dial to AC voltage (

\text{V}\sim

).

Measuring potential difference requires configuring the digital multimeter as a voltmeter by placing the red probe into the

\text{V}/\Omega

terminal and the black probe into

\text{COM}

, selecting AC voltage (

\text{V}\sim

), and connecting the probes in parallel across the energized component. A voltmeter has very high internal impedance so that it draws negligible current and does not disturb circuit operation.

Step-by-Step Solution

1

Determine the measurement type and power source characteristics

The target measurement is electrical potential difference (voltage) on a

120\text{ V}

alternating current (AC) power source.

Multimeter function selection requires knowing whether the signal is AC or DC and whether voltage, current, or resistance is being evaluated.

2

Select the proper multimeter input jacks and dial position

Black lead in

\text{COM}

, red lead in

\text{V}/\Omega

, and selector dial switched to AC Volts (

\text{V}\sim

).

The

\text{V}/\Omega

port routes the signal through high-impedance circuitry, and the AC voltage mode properly samples and computes the root-mean-square (RMS) voltage.

3

Identify the proper circuit connection topology

Connect test leads across (in parallel with) the fan motor terminals while the circuit is live.

Voltmeters possess extremely high internal resistance (typically

>10\text{ M}\Omega

) and must be placed across a component to measure potential difference without altering circuit current.

Key Concept

Multimeter configuration for AC voltage measurement requires parallel connection across the load and high-impedance voltage jack selection.

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