All practice questions

Question 3261Question →

A robotic transport carrier with a mass of $200\text{ kg}$ starts from rest and accelerates uniformly along a straight track, reaching a final velocity of $15\text{ m/s}$ in $5\text{ s}$ . What is the magnitude of the net force acting on the carrier?

60\text{ N}

600\text{ N}

6{}000\text{ N}

300\text{ N}

Show answer & explanation

Answer: $600\text{ N}$

Answer

600\text{ N}

The correct answer is

600\text{ N}

. First, find the constant acceleration by dividing the change in velocity by elapsed time:

a = \frac{15\text{ m/s}}{5\text{ s}} = 3\text{ m/s}^2

. Next, use Newton's second law of motion (

F = ma

) to find the net force:

F = 200\text{ kg} \times 3\text{ m/s}^2 = 600\text{ N}

.

Step-by-Step Solution

1

Calculate the linear acceleration using the kinematic formula

a = \frac{\Delta v}{\Delta t}

.

a = \frac{15\text{ m/s} - 0\text{ m/s}}{5\text{ s}} = 3\text{ m/s}^2

Acceleration represents the rate at which velocity changes over time.

2

Apply Newton's second law of motion (

F_{\text{net}} = m \times a

).

F_{\text{net}} = 200\text{ kg} \times 3\text{ m/s}^2 = 600\text{ N}

The net force required to accelerate an object is equal to the product of its mass and its acceleration.

Key Concept

Newton's Second Law of Motion and Kinematic Acceleration

Estimated Time:45s

Question 3262Question →

An automated wire-cutting machine processes $350$ meters of copper wiring in $20$ minutes. Operating at this constant rate, how many meters of wiring will the machine process in $1.5$ hours?

Show answer & explanation

Answer: 1575

Answer

1575

Because the machine operates at a constant rate, the relationship between time and wire processed is directly proportional. However, the time units differ between the given rate (

20

minutes) and the target time (

1.5

hours). First, convert

1.5

hours to

90

minutes. Then, set up the direct proportion

\frac{350 \text{ meters}}{20 \text{ minutes}} = \frac{x \text{ meters}}{90 \text{ minutes}}

. Solving for

x

by multiplying

350

by

90

and dividing by

20

yields exactly

1575

meters.

Step-by-Step Solution

1

Harmonize the time units by converting hours to minutes.

90

minutes

The given rate is in meters per minute, so the target time must also be in minutes to establish a valid equivalent ratio.

2

Set up a direct proportion equation.

\frac{350}{20} = \frac{x}{90}

The length of wire processed increases at a constant, directly proportional rate to the time operated.

3

Solve the proportion for the unknown value

x

.

x = 1575

Cross-multiplying and simplifying isolates

x

, calculating the total meters processed in the longer timeframe.

Key Concept

Direct proportions with unit harmonization

Question 3263Question →

An open pot of pure water on a stove reaches a vigorous boil at standard atmospheric pressure. While heat continues to be supplied by the burner, which of the following describes what happens to the temperature of the boiling liquid water as it turns to steam?

The temperature steadily rises above

100^\circ\text{C}

as the liquid absorbs additional thermal energy.

The temperature remains constant at

100^\circ\text{C}

because the added thermal energy is used to break intermolecular bonds during the phase change.

The temperature rapidly drops toward

0^\circ\text{C}

because escaping steam carries away all thermal energy from the remaining liquid.

The temperature fluctuates unpredictably between the freezing point and the boiling point due to turbulent convection currents.

Show answer & explanation

Answer: The temperature remains constant at $100^\circ\text{C}$ because the added thermal energy is used to break intermolecular bonds during the phase change.

Answer

The temperature remains constant at

100^\circ\text{C}

because the added thermal energy is used to break intermolecular bonds during the phase change.

When a pure substance reaches its boiling point at constant pressure, its temperature remains unchanged until the entire liquid has transitioned into vapor. The continuous input of heat supplies the latent heat of vaporization, which breaks the intermolecular bonds holding liquid molecules together rather than increasing their average kinetic energy.

Step-by-Step Solution

1

Identify the physical state and process occurring in the system.

Water is actively undergoing a phase change from liquid to gas (boiling/vaporization) at standard atmospheric pressure (

1\text{ atm}

).

Phase transitions have specific thermodynamic properties distinct from single-phase heating.

2

Analyze how thermal energy affects substances during a phase transition.

The added energy functions as the latent heat of vaporization, which overcomes intermolecular hydrogen bonds rather than increasing molecular kinetic energy.

Temperature is a measure of average translational kinetic energy; because the kinetic energy does not increase during the phase change, temperature remains constant until all liquid has vaporized.

Key Concept

Latent heat and constant temperature during phase transitions

Estimated Time:1m 0s

Question 3264Question →

A plumber is installing copper piping and starts with a full pipe that measures $14 \frac{1}{8}$ feet in length. For the installation, he cuts off one section measuring $3 \frac{1}{2}$ feet and another section measuring $5 \frac{3}{8}$ feet. Assuming no material is lost to the saw blade during the cuts, what is the length of the remaining pipe? Enter your answer as a decimal.

Show answer & explanation

Answer: 5.25

Answer

The length of the remaining pipe is 5.25 feet.

First, the total length of the cut sections is found by getting a common denominator (

3 \frac{4}{8} + 5 \frac{3}{8} = 8 \frac{7}{8}

feet). To find the remaining pipe, this total is subtracted from the original length. Since

\frac{7}{8}

is larger than

\frac{1}{8}

, regrouping is necessary:

14 \frac{1}{8}

becomes

13 \frac{9}{8}

. Subtracting yields

5 \frac{2}{8}

, which simplifies to

5 \frac{1}{4}

. The decimal equivalent of

5 \frac{1}{4}

is 5.25.

Step-by-Step Solution

1

Find a common denominator for the two sections that were cut.

3 \frac{1}{2} = 3 \frac{4}{8}

Fractions must have the same denominator before they can be added.

2

Add the lengths of the two cut sections together.

3 \frac{4}{8} + 5 \frac{3}{8} = 8 \frac{7}{8}

feet

This determines the total amount of pipe removed from the original length.

3

Set up the subtraction to find the remaining pipe.

14 \frac{1}{8} - 8 \frac{7}{8}

Subtracting the removed amount from the original total gives the remaining balance.

4

Regroup the initial total to allow for fraction subtraction.

14 \frac{1}{8} = 13 + 1 \frac{1}{8} = 13 \frac{9}{8}

You cannot subtract

\frac{7}{8}

from

\frac{1}{8}

without borrowing a whole number (1 =

\frac{8}{8}

).

5

Subtract the whole numbers and the fractions.

13 \frac{9}{8} - 8 \frac{7}{8} = 5 \frac{2}{8}

feet, which simplifies to

5 \frac{1}{4}

feet.

This isolates the precise length of the pipe that is left.

6

Convert the simplified fraction into a decimal format.

5 \frac{1}{4} = 5.25

The question specifically requests the final answer to be entered as a decimal.

Key Concept

Fraction Addition and Subtraction in Context

Question 3265Question →

A technician observes several maintenance procedures being performed in an electronics repair depot. Which of the following processes represents a chemical change?

Melting solder wire to join two electrical conductors

Tarnishing of silver electrical contacts upon exposure to airborne sulfur

Evaporating isopropyl alcohol solvent from a cleaned circuit board

Bending a copper grounding strap to fit inside an equipment chassis

Show answer & explanation

Answer: Tarnishing of silver electrical contacts upon exposure to airborne sulfur

Answer

Tarnishing of silver electrical contacts upon exposure to airborne sulfur

Tarnishing of silver contacts is a chemical change because silver reacts chemically with sulfur in the environment to produce silver sulfide (

Ag_2S

), creating a new chemical compound with distinct physical and chemical properties.

Step-by-Step Solution

1

Differentiate between physical and chemical changes in matter

A physical change alters form, shape, or state of matter without changing chemical composition, whereas a chemical change breaks or forms chemical bonds to produce one or more new substances.

Establishing the core scientific criteria allows systematic evaluation of each scenario.

2

Classify the molecular behavior in each observed procedure

Melting solder and evaporating alcohol are phase changes (solid to liquid, liquid to gas), and bending copper is mechanical deformation—all physical changes. Tarnishing involves silver atoms chemically combining with sulfur to yield silver sulfide (

Ag_2S

), a new substance.

Identifying which process produces a chemically distinct substance isolates the chemical change.

Key Concept

Distinction between physical changes (phase and shape changes) and chemical changes (formation of new chemical substances)

Estimated Time:45s

Question 3266Question →

Thermal management systems and physical phenomena rely on distinct thermodynamic processes to transfer or absorb energy. Match each thermal phenomenon or device on the left with the primary thermodynamic principle that explains its operation on the right.

Click a left item, then click its matching right item

Show answer & explanation

Answer

Aerosol canister cooling matches adiabatic expansion doing work and decreasing internal energy; Cooling tower matches absorption of latent heat of vaporization during liquid-to-gas phase change; Evacuated double-wall flask matches elimination of a material medium to stop conduction and convection; Aluminum finned heat sink matches increasing surface area to maximize convective and radiative dissipation.

Each application precisely maps to its foundational thermodynamic mechanism: aerosol discharge is an adiabatic expansion lowering temperature; cooling towers utilize latent heat of vaporization; vacuum flasks eliminate the physical medium to block conduction and convection; and finned heat sinks expand surface area to enhance fluid convection.

Step-by-Step Solution

1

Analyze the aerosol canister phenomenon

Identified as adiabatic gas expansion

Rapid discharge does not allow heat transfer from the environment during expansion, causing internal kinetic energy to decrease as work is performed.

2

Analyze the cooling tower operation

Identified as phase-change latent heat absorption

Water evaporation requires high latent heat of vaporization, extracting thermal energy from the remaining water volume.

3

Analyze the vacuum flask structure

Identified as medium elimination to block conduction and convection

Thermal conduction and convection depend entirely on molecular interactions within a material medium, which is absent in a vacuum.

4

Analyze the finned heat sink geometry

Identified as geometric surface area expansion for convection

The rate of heat transfer by convection is directly proportional to the surface area in contact with the cooling fluid.

Key Concept

Thermodynamic heat transfer mechanisms (conduction, convection, radiation), phase change latent heat, and adiabatic gas processes.

Estimated Time:1m 15s

Question 3267Question →

An industrial supply company purchases a heavy-duty welding machine directly from the manufacturer for $\$2,400.00$ . The company applies a $40\%$ markup to establish their standard retail price. During a promotional event, a customer purchases the machine at a $15\%$ discount off the retail price. If an additional $6\%$ sales tax is applied to the discounted price, what is the total amount the customer pays in dollars?

Show answer & explanation

Answer: 3027.36

Answer

3027.36

To find the final amount, each percentage change must be applied sequentially to the most recent price. First, the

40\%

markup increases the

\$2,400.00

base price to

\$3,360.00

. Next, the

15\%

discount reduces this retail price to

\$2,856.00

. Finally, applying the

6\%

sales tax to the discounted price yields a final total of

\$3,027.36

.

Step-by-Step Solution

1

Calculate the markup amount and the standard retail price.

\$2,400.00 \times 1.40 = \$3,360.00

The

40\%

markup is based on the initial wholesale cost.

2

Calculate the sale price after the discount.

\$3,360.00 \times (1 - 0.15) = \$3,360.00 \times 0.85 = \$2,856.00

The

15\%

discount is taken off the standard retail price, not the original cost.

3

Calculate the final price including sales tax.

\$2,856.00 \times 1.06 = \$3,027.36

Sales tax is calculated on the discounted sale price to find the total amount paid.

Key Concept

Calculating successive percentage changes (markups, discounts, and taxes) by updating the base value at each step.

Question 3268Question →

A forward operating base maintains a stockpile of ammunition, food, and medical supplies in a weight ratio of 5:3:2. An incoming supply convoy delivers an additional 3 tons of food and 8 tons of medical supplies, while the ammunition stockpile remains unchanged. After the delivery, the new weight ratio of food to medical supplies becomes 4:3. What was the total weight of the entire stockpile before the convoy arrived?

20 tons

23 tons

115 tons

230 tons

Show answer & explanation

Answer: 230 tons

Answer

The total weight of the entire stockpile before the convoy arrived was 230 tons.

The original weights can be expressed proportionally as

5x

(ammunition),

3x

(food), and

2x

(medical supplies), totaling

10x

. After adding the newly delivered amounts, the new weights for food and medical supplies become

3x + 3

and

2x + 8

. Setting their ratio equal to 4:3 yields the equation

\frac{3x + 3}{2x + 8} = \frac{4}{3}

. Cross-multiplying gives

9x + 9 = 8x + 32

, which simplifies to

x = 23

. The total original weight is therefore

10x = 10(23) = 230

tons.

Step-by-Step Solution

1

Define the original weights of the supplies using a common multiplier

x

.

Ammunition =

5x

, Food =

3x

, Medical =

2x

. Total original weight =

5x + 3x + 2x = 10x

.

A ratio represents proportional parts of a whole. Multiplying each ratio term by a constant

x

gives expressions for their actual quantities.

2

Set up an algebraic expression for the new weights of food and medical supplies after the delivery.

New Food =

3x + 3

; New Medical =

2x + 8

.

The convoy delivered 3 tons of food and 8 tons of medical supplies, which are added to their respective original expressions.

3

Create a proportion using the new ratio of food to medical supplies provided in the problem.

\frac{3x + 3}{2x + 8} = \frac{4}{3}

The problem states that after the delivery, the ratio of food to medical supplies becomes 4:3.

4

Solve the proportion for

x

using cross-multiplication.

3(3x + 3) = 4(2x + 8) \Rightarrow 9x + 9 = 8x + 32 \Rightarrow x = 23

.

Cross-multiplying eliminates the fractions, creating a linear equation that can be solved for the common multiplier

x

.

5

Calculate the total original weight of the stockpile by substituting the value of

x

.

10(23) = 230

tons.

The question asks for the total weight of the entire stockpile before the delivery, which is represented by the expression

10x

.

Key Concept

Setting up and solving algebraic proportions from multi-term ratios after quantities undergo additive changes.

Estimated Time:1m 30s

Question 3269Question →

A diner's meal costs $50.00. If the diner leaves a 20% tip, what is the total amount paid?

$10.00

$51.00

$60.00

$70.00

Show answer & explanation

Answer: $60.00

Answer

$60.00

The correct total is found by converting 20% to a decimal (0.20), multiplying it by

50.00 to find the tip amount (

10.00), and then adding that tip to the original

50.00 bill to get a total of

60.00.

Step-by-Step Solution

1

Calculate the amount of the tip by finding 20% of the meal cost ($50.00).

$10.00

The tip is a percentage based on the initial cost of the meal.

2

Add the tip amount to the original meal cost.

$60.00

The total amount paid includes both the original meal cost and the tip.

Key Concept

Calculating a percentage of a quantity and adding it to the base amount.

Question 3270Question →

A field water purification system processes $12.5$ gallons of clean water every $5$ minutes. Operating at this same constant rate, how many gallons of clean water will the system process in $2$ hours?

5

30

48

300

Show answer & explanation

Answer: 300

Answer

The system will process 300 gallons of clean water in 2 hours.

The correct answer accurately harmonizes the time units by converting 2 hours into 120 minutes, and then scales the given volume-to-time ratio (

12.5

gallons per

5

minutes) up to

120

minutes, resulting in

300

gallons.

Step-by-Step Solution

1

Convert the target time into the same units used in the given rate.

2

hours

= 2 \times 60 = 120

minutes.

Proportions require matching units to accurately compare the two ratios.

2

Calculate the unit rate of water processed per minute.

12.5 \text{ gallons} \div 5 \text{ minutes} = 2.5 \text{ gallons per minute}

.

Finding the amount processed in one minute simplifies scaling up to the total time.

3

Multiply the unit rate by the total number of minutes.

2.5 \text{ gallons/minute} \times 120 \text{ minutes} = 300 \text{ gallons}

.

This applies the constant rate over the entire duration to find the total volume.

Key Concept

Direct proportions with unit conversion

Question 3271Question →

A specialized construction $3$ D printer uses $2.4$ metric tons of concrete to print $15$ linear feet of a wall. At this constant rate, how many kilograms of concrete are required to print a wall that is $25$ yards long? ( $1$ metric ton = $1,000$ kilograms; $1$ yard = $3$ feet)

4,000

1,200

12,000

480

Show answer & explanation

Answer: $12,000$

Answer

The correct answer is

12,000

.

The correct answer is found by harmonizing the length units, determining the scaled proportion, and harmonizing the final weight units. Converting

25

yards to

75

feet reveals that the target length is

5

times the initial length (

75 \div 15 = 5

). Multiplying the initial concrete requirement by this scale factor yields

12

metric tons (

2.4 \times 5 = 12

). Finally, converting

12

metric tons to kilograms yields

12,000

kilograms.

Step-by-Step Solution

1

Convert the target length from yards to feet.

25 \text{ yards} = 25 \times 3 = 75 \text{ feet}

.

The units of length must match the given rate (

15

linear feet) before setting up a proportion.

2

Set up a direct proportion to find the metric tons of concrete needed.

\frac{2.4}{15} = \frac{x}{75}

The amount of concrete used is directly proportional to the length of the wall printed.

3

Solve the proportion for

x

.

x = \frac{2.4 \times 75}{15} = 2.4 \times 5 = 12 \text{ metric tons}

.

Isolating

x

provides the required amount of concrete in metric tons.

4

Convert metric tons to kilograms.

12 \text{ metric tons} = 12 \times 1,000 = 12,000 \text{ kilograms}

.

The question specifically asks for the final answer in kilograms.

Key Concept

Direct Proportions with Unit Conversions

Question 3272Question →

An automotive cooling system uses a water pump to continuously circulate liquid coolant through channels in the engine block, carrying thermal energy away from hot metal components to the radiator. Which heat transfer mechanism is primarily responsible for transporting thermal energy throughout the moving liquid?

Conduction

Convection

Radiation

Sublimation

Show answer & explanation

Answer: Convection

Answer

Convection

Convection is the transfer of thermal energy by the bulk movement and circulation of a fluid (either liquid or gas). When the water pump circulates liquid coolant through the engine and out to the radiator, the flowing liquid physically carries the thermal energy with it.

Step-by-Step Solution

1

Identify the physical medium and method of transport in the scenario.

Thermal energy is carried away by the physical, macroscopic circulation of a liquid fluid (coolant).

Identifying how the matter carrying the heat moves determines the fundamental heat transfer mechanism.

2

Classify the transport mechanism among standard thermodynamic processes.

Heat transfer via the bulk motion and circulation of fluids is classified as convection.

Conduction occurs via stationary molecular collisions, radiation occurs via electromagnetic waves, and convection occurs through fluid movement.

Key Concept

Convection is the transfer of thermal energy through the bulk movement and circulation of fluids (liquids and gases).

Estimated Time:1m 0s

Question 3273Question →

Matter exhibits distinct measurable characteristics that determine how it behaves under mechanical forces, thermal energy, and phase equilibria. Match each physical property of matter on the left with its corresponding scientific description on the right.

Click a left item, then click its matching right item

Show answer & explanation

Answer

Hardness matches with the resistance of a solid substance to surface scratching, localized plastic deformation, or mechanical abrasion; Specific Heat Capacity matches with the quantity of thermal energy required to raise the temperature of one unit mass of a substance by one degree Celsius; Elasticity matches with the capacity of a deformed solid material to return to its original shape and size once applied deforming forces are removed; Vapor Pressure matches with the force per unit area exerted by an evaporated gas in thermodynamic equilibrium with its condensed liquid or solid phase at a given temperature.

Each physical property is correctly paired with its foundational scientific definition: Hardness pertains to surface scratch resistance, Specific Heat Capacity defines the heat required to change temperature per unit mass, Elasticity governs reversible mechanical deformation, and Vapor Pressure quantifies phase equilibrium pressure in a closed container.

Step-by-Step Solution

1

Analyze the definition of Hardness

Hardness corresponds to resistance to scratching, penetration, or abrasion on a solid surface.

Hardness is evaluated on scales like the Mohs hardness scale by testing which substance can scratch another.

2

Analyze the definition of Specific Heat Capacity

Specific Heat Capacity corresponds to the amount of heat energy per unit mass needed to raise temperature by

1^\circ\text{C}

.

Thermal properties govern how substances absorb and transfer heat without undergoing chemical decomposition.

3

Analyze the definition of Elasticity

Elasticity corresponds to the ability of a material to rebound to its original shape after deforming forces are removed.

Mechanical elasticity is characterized by Hooke's Law within the proportional limit of the material.

4

Analyze the definition of Vapor Pressure

Vapor Pressure corresponds to the equilibrium pressure created by gas particles above a liquid or solid phase in a closed system.

Volatile substances with weak intermolecular forces display higher vapor pressures at a given temperature.

Key Concept

Physical properties of matter describe observable or measurable characteristics of substances that do not alter their underlying chemical composition.

Estimated Time:1m 0s

Question 3274Question →

Household refrigerators transfer thermal energy from a cooler interior to a warmer room through a continuous closed-loop thermodynamic cycle. Starting with the stage where low-pressure refrigerant vapor enters the compressor, what is the correct sequential order of these stages in the refrigeration cycle?

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The compressor pressurizes and heats the vapor, followed by the condenser releasing heat to turn the vapor into liquid, then the expansion valve lowering the pressure and temperature, and finally the evaporator absorbing heat from the interior compartment as the liquid vaporizes.

The correct sequence follows the standard vapor-compression cycle: compression raises the temperature and pressure of the vapor, condensation expels heat to the environment to form a liquid, expansion drops the pressure and chills the liquid, and evaporation absorbs heat from the cold space to turn the fluid back into vapor.

Step-by-Step Solution

1

Identify the initial compression stage.

The compressor does mechanical work to compress the low-pressure vapor into a high-pressure, high-temperature gas.

Raising the pressure and temperature ensures the refrigerant is hotter than the ambient room air.

2

Follow the refrigerant into the exterior condenser coils.

Heat transfers from the hot refrigerant to the surrounding room, causing the refrigerant to condense into a high-pressure liquid.

Thermal energy naturally transfers from the hotter gas to the cooler room air.

3

Track the fluid through the expansion valve.

The high-pressure liquid passes through an expansion valve, where pressure suddenly drops, causing a steep decline in temperature.

Rapid fluid expansion and pressure reduction dramatically lower the boiling point and temperature of the refrigerant.

4

Analyze heat absorption in the interior evaporator coils.

The cold refrigerant absorbs thermal energy from the refrigerator interior and boils into a low-pressure gas.

Because the refrigerant is now colder than the interior air, heat transfers into the fluid, cooling the compartment.

Key Concept

Thermodynamic refrigeration cycle, phase transitions, and forced heat transfer

Estimated Time:1m 0s

Question 3275Question →

In classical mechanics, specific physical quantities and principles describe how bodies move and respond to applied forces. Match each mechanical concept on the left with its correct physical definition on the right.

Click a left item, then click its matching right item

Show answer & explanation

Answer

Inertia matches the natural tendency of an object to resist any change in its state of rest or motion; Acceleration matches the rate at which an object changes its velocity with respect to time; Linear Momentum matches the product of an object's mass and its velocity; Centripetal Force matches a net inward force directed toward the center of curvature that maintains circular motion.

Inertia is the tendency of matter to resist changes in motion; acceleration is the time derivative/rate of change of velocity; linear momentum represents mass in motion (

p = mv

); and centripetal force is the inward radial force that continuously changes the direction of an object in circular motion.

Step-by-Step Solution

1

Identify the definition of Inertia

Inertia corresponds to the property of mass resisting alterations in its state of rest or uniform motion.

According to Newton's First Law of Motion, inertia is an intrinsic resistance to acceleration.

2

Identify the definition of Acceleration

Acceleration corresponds to the rate of change of velocity over time (

a = \frac{\Delta v}{\Delta t}

).

Acceleration describes how rapidly speed or direction changes over a given time interval.

3

Identify the definition of Linear Momentum

Linear Momentum corresponds to mass multiplied by velocity (

p = mv

).

Momentum quantifies the motion of a mass and is conserved in isolated systems.

4

Identify the definition of Centripetal Force

Centripetal Force corresponds to the center-seeking net force in circular motion (

F_c = \frac{mv^2}{r}

).

An inward radial force is required to continuously deflect an object's velocity vector along a curve.

Key Concept

Fundamental definitions and kinematic relationships governing linear and circular mechanics.

Estimated Time:1m 0s

Question 3276Question →

A culinary school is purchasing a set of commercial stand mixers. The wholesale cost of the equipment is $\$1,500.00$ . A restaurant supply store applies a $30\%$ markup to the wholesale cost to determine the regular retail price. To support the culinary program, the store then applies a $15\%$ educational discount to this regular retail price. What is the final price the school pays for the mixers before any applicable taxes?

\$1,657.50

\$1,725.00

\$292.50

\$1,935.00

Show answer & explanation

Answer: $\$1,657.50$

Answer

\$1,657.50

To solve this problem, first determine the retail price by calculating the

30\%

markup on the

\$1,500.00

wholesale cost. The markup is

\$450.00

, making the retail price

\$1,950.00

. Then, calculate the

15\%

educational discount based on this new retail price. The discount is

\$292.50

(

0.15 \times \$1,950.00

). Subtracting this discount from the retail price gives a final cost of

\$1,657.50

.

Step-by-Step Solution

1

Calculate the markup amount by finding

30\%

of the wholesale cost.

\$1,500.00 \times 0.30 = \$450.00

The store increases the wholesale price by

30\%

to determine its retail price.

2

Add the markup amount to the original wholesale cost to find the regular retail price.

\$1,500.00 + \$450.00 = \$1,950.00

The retail price is the sum of the wholesale cost and the markup.

3

Calculate the discount amount by finding

15\%

of the regular retail price.

\$1,950.00 \times 0.15 = \$292.50

The educational discount is applied to the store's regular retail price, not the wholesale price.

4

Subtract the discount amount from the retail price to find the final price.

\$1,950.00 - \$292.50 = \$1,657.50

The final amount paid is the retail price minus the calculated discount.

Key Concept

Successive Percentage Changes (Markup followed by Discount)

Question 3277Question →

Read the scenario and fill in the missing numerical values.

Fill in the blanks below

A customer gets a haircut that costs

\$20.00

. If the customer leaves a

15\%

tip, the tip amount is $\$

and the total cost of the haircut is

\$$.

Show answer & explanation

Answer

The tip amount is

3.00 and the total cost is

23.00.

The tip amount is found by multiplying the base cost by the tip percentage expressed as a decimal (

\$20.00 \times 0.15 = \$3.00

). The total cost is determined by adding this calculated tip amount to the original price (

\$20.00 + \$3.00 = \$23.00

).

Step-by-Step Solution

1

Convert the tip percentage to a decimal and multiply by the cost of the haircut.

\$20.00 \times 0.15 = \$3.00

To find the dollar amount of the tip, calculate

15\%

of the original price.

2

Add the calculated tip amount to the original cost.

\$20.00 + \$3.00 = \$23.00

The total cost is the sum of the original service price and the tip.

Key Concept

Calculating tip amounts using percentages and finding the final total cost.

Question 3278Question →

An astronaut collects a rock sample on the Moon and measures its physical properties before transporting it back to Earth. If the gravitational pull on the surface of the Moon is approximately one-sixth of Earth's gravity, how do the mass and weight of the rock sample on the Moon compare to its mass and weight on Earth?

The sample has the same weight on the Moon as on Earth, but its mass is one-sixth of its Earth mass.

The sample has the same mass on the Moon as on Earth, but its weight is one-sixth of its Earth weight.

Both the mass and the weight of the sample are one-sixth of their respective values on Earth.

Both the mass and the weight of the sample remain identical in both locations.

Show answer & explanation

Answer: The sample has the same mass on the Moon as on Earth, but its weight is one-sixth of its Earth weight.

Answer

The sample has the same mass on the Moon as on Earth, but its weight is one-sixth of its Earth weight.

Mass is an intrinsic property of matter that quantifies how much substance an object contains, meaning it remains identical regardless of where the object is located in the universe. In contrast, weight is the gravitational force acting on that mass, expressed as

W = mg

. Because gravity on the surface of the Moon is approximately one-sixth that of Earth, the rock sample weighs one-sixth as much on the Moon while retaining its full mass.

Step-by-Step Solution

1

Define mass in the context of physical properties of matter.

Mass represents the fundamental quantity of matter contained within an object and is invariant regardless of location or external forces.

Mass does not depend on gravitational acceleration.

2

Define weight and determine how local gravity affects it.

Weight is the force exerted on an object by gravity, given by the relationship

W = mg

. Since lunar gravity is

\frac{1}{6}

of Earth's gravity, the weight of the rock on the Moon is

\frac{1}{6}

of its weight on Earth.

Weight is a force that varies directly with local gravitational field strength.

3

Synthesize the behavior of both properties.

The rock maintains an identical mass across both environments, while its weight on the Moon is one-sixth of its weight on Earth.

Combining an invariant scalar mass with a variable gravitational force yields the correct comparison.

Key Concept

Mass versus Weight in Gravitational Fields

Question 3279Question →

A military water purification unit has three filters operating at different unit rates. Filter X can purify $180$ gallons of water in $1.5$ hours. Filter Y can purify $320$ gallons in $2$ hours. Filter Z's purification rate is exactly $20\%$ faster than Filter X's rate. If all three filters are turned on at the same time, how many minutes will it take them to completely purify $1,484$ gallons of water?

Show answer & explanation

Answer: 210

Answer

210

The correct answer is found by first determining the individual hourly rates of each filter, adding them together for a combined rate, calculating the total hours required for the given volume, and finally converting that time into minutes.

Step-by-Step Solution

1

Calculate the unit rate for Filter X.

120

gallons per hour

To find the combined rate later, we first need the individual hourly rate of Filter X.

180 \div 1.5 = 120

.

2

Calculate the unit rate for Filter Y.

160

gallons per hour

We need the individual hourly rate of Filter Y to find the total combined rate.

320 \div 2 = 160

.

3

Calculate the unit rate for Filter Z.

144

gallons per hour

Filter Z operates

20\%

faster than Filter X, so its rate is

120 + (0.20 \times 120) = 144

.

4

Calculate the combined unit rate of all three filters.

424

gallons per hour

Since the filters operate simultaneously, their individual rates add together:

120 + 160 + 144 = 424

.

5

Determine the time in hours needed to purify

1,484

gallons.

3.5

hours

Time equals the total amount required divided by the combined rate:

1,484 \div 424 = 3.5

.

6

Convert the total time from hours to minutes.

210

minutes

The question explicitly asks for the answer in minutes.

3.5 \times 60 = 210

.

Key Concept

Calculating, modifying, and combining multiple unit rates, followed by a final unit conversion.

Alternative Method

You could convert the rates to gallons per minute immediately. Filter X is 2 gallons per minute, Filter Y is 2.67 gallons per minute, etc. However, this introduces repeating decimals early on, making the math significantly harder without a calculator. Keeping the rates in hours avoids complex fractions until the final step.

Estimated Time:3m 30s

Question 3280Question →

A combat engineer is preparing a concrete mixture for a runway repair. The project requires a total of $18 \frac{1}{4}$ cubic feet of sand. The engineer has two available piles of sand: one containing $7 \frac{4}{5}$ cubic feet and another containing $5 \frac{1}{2}$ cubic feet. If both piles are combined, how many more cubic feet of sand are needed to meet the total requirement?

5 \frac{15}{28}

5 \frac{1}{20}

4 \frac{19}{20}

5 \frac{19}{20}

Show answer & explanation

Answer: $4 \frac{19}{20}$

Answer

The correct amount of additional sand needed is

4 \frac{19}{20}

cubic feet.

The correct calculation requires first summing the available resources (

7 \frac{4}{5} + 5 \frac{1}{2} = 13 \frac{3}{10}

) and then subtracting this from the target amount (

18 \frac{1}{4} - 13 \frac{3}{10}

). Converting to a common denominator of

20

and regrouping the whole number correctly yields exactly

4 \frac{19}{20}

.

Step-by-Step Solution

1

Calculate the total volume of available sand by adding the two piles together.

7 \frac{4}{5} + 5 \frac{1}{2} = 7 \frac{8}{10} + 5 \frac{5}{10} = 12 \frac{13}{10}

. Since

\frac{13}{10}

is an improper fraction (

1 \frac{3}{10}

), this simplifies to

13 \frac{3}{10}

cubic feet.

You must first determine how much sand is already on hand before calculating the remaining amount needed.

2

Find a common denominator to subtract the available sand from the total required sand.

The least common denominator for

4

and

10

is

20

. Convert the fractions:

18 \frac{1}{4}

becomes

18 \frac{5}{20}

, and

13 \frac{3}{10}

becomes

13 \frac{6}{20}

.

Fractions must have equivalent denominators before they can be added or subtracted.

3

Subtract the available sand from the required total by regrouping (borrowing).

Subtract

13 \frac{6}{20}

from

18 \frac{5}{20}

. Since

5

is less than

6

, borrow

1

from the whole number

18

to get

17 \frac{25}{20}

. Then,

17 \frac{25}{20} - 13 \frac{6}{20} = 4 \frac{19}{20}

cubic feet.

Regrouping is necessary when the fraction being subtracted is larger than the fraction it is being subtracted from.

Key Concept

Adding and subtracting mixed numbers with unlike denominators, including regrouping (borrowing).

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