Unit Conversions Practice Questions — SAT

Question 1Question →

A laboratory dispensing pump releases a liquid solution at a constant rate of $0.75\text{ deciliters per minute}$ . If $1\text{ deciliter} = 100\text{ milliliters}$ , what is the pump's dispensing rate, in milliliters per second?

Show answer & explanation

Answer: 1.25

Answer

The correct answer is 1.25. The pump dispenses 1.25 milliliters per second.

To find the dispensing rate in milliliters per second, we first convert the volume from deciliters to milliliters. Multiplying

0.75\text{ deciliters}

by

100\text{ milliliters per deciliter}

gives

75\text{ milliliters per minute}

. Next, to convert minutes to seconds, we divide the rate by

60

(since

1\text{ minute} = 60\text{ seconds}

), which yields

\frac{75}{60} = 1.25\text{ milliliters per second}

.

Step-by-Step Solution

1

Convert the dispensing rate from deciliters per minute to milliliters per minute using the conversion factor

1\text{ deciliter} = 100\text{ milliliters}

.

0.75\text{ deciliters/minute} \times 100\text{ milliliters/deciliter} = 75\text{ milliliters/minute}

This converts the volume unit from deciliters to milliliters.

2

Convert the rate from milliliters per minute to milliliters per second by dividing by

60

, since there are

60\text{ seconds}

in

1\text{ minute}

.

75\text{ milliliters/minute} \div 60\text{ seconds/minute} = 1.25\text{ milliliters/second}

This converts the time unit from minutes to seconds, yielding the final rate.

Key Concept

Unit Conversions

Alternative Method

Alternatively, you can convert the units in a single dimensional analysis step:

0.75 \text{ dL/min} \times \frac{100 \text{ mL}}{1 \text{ dL}} \times \frac{1 \text{ min}}{60 \text{ s}} = \frac{75}{60} \text{ mL/s} = 1.25 \text{ mL/s}

.

Estimated Time:1m 30s

Question 2Question →

A toy train moves along a track at a constant rate of $15$ feet per second. What is the train's speed, in yards per second? (Given that $1\text{ yard} = 3\text{ feet}$ )

5

12

18

45

Show answer & explanation

Answer: 5

Answer

5

The correct answer is 5. To convert the speed from feet per second to yards per second, divide the rate of

15

feet per second by the conversion factor of

3

feet per yard. This yields

5

yards per second.

Step-by-Step Solution

1

Identify the given rate and the conversion factor.

The given rate is

15\text{ feet per second}

, and the conversion factor is

1\text{ yard} = 3\text{ feet}

.

This establishes the relationship between the current unit (feet) and the target unit (yards).

2

Convert the speed by dividing by the conversion factor.

\frac{15\text{ feet}}{1\text{ second}} \times \frac{1\text{ yard}}{3\text{ feet}} = 5\text{ yards per second}

.

Dividing the speed in feet per second by 3 converts the distance unit from feet to yards, resulting in the correct rate of 5 yards per second.

Key Concept

Unit Conversions

Estimated Time:45s

Question 3Question →

A garden hose discharges water at a constant rate of $8$ quarts per minute. What is this rate, in gallons per hour? (Given that $1\text{ gallon} = 4\text{ quarts}$ )

Show answer & explanation

Answer: 120

Answer

120

To convert the discharge rate from quarts per minute to gallons per hour, we first convert quarts to gallons. Since

1\text{ gallon} = 4\text{ quarts}

, we divide the rate of

8

quarts per minute by

4

to get

2

gallons per minute. Next, to convert minutes to hours, we multiply this rate by

60

(since there are

60

minutes in

1\text{ hour}

). This gives

2 \times 60 = 120

gallons per hour.

Step-by-Step Solution

1

Convert quarts per minute to gallons per minute

2

gallons per minute

Since

1\text{ gallon} = 4\text{ quarts}

, divide the flow rate of

8

quarts per minute by the conversion factor of

4

to find the rate in gallons per minute:

\frac{8\text{ quarts/min}}{4\text{ quarts/gallon}} = 2\text{ gallons/min}

.

2

Convert gallons per minute to gallons per hour

120

gallons per hour

Since there are

60

minutes in an hour, multiply the rate of

2

gallons per minute by

60

to find the total gallons discharged in one hour:

2\text{ gallons/min} \times 60\text{ min/hour} = 120\text{ gallons/hour}

.

Key Concept

Unit Conversions

Question 4Question →

A 3D printer extrudes plastic filament at a constant rate of $8$ millimeters per second. What is this rate, in meters per hour?

0.48

2.88

28.8

28,800

Show answer & explanation

Answer: $28.8$

Answer

The rate is

28.8

meters per hour.

To convert the rate from millimeters per second to meters per hour, we apply the unit conversion factors. We convert millimeters to meters by dividing by

1000

, and convert seconds to hours by multiplying by

3600

. This gives:

8\text{ mm/s} \times \frac{1\text{ m}}{1000\text{ mm}} \times \frac{3600\text{ s}}{1\text{ hour}} = 8 \times 3.6 = 28.8\text{ meters per hour}

.

Step-by-Step Solution

1

Convert the length unit from millimeters to meters.

Since

1\text{ meter} = 1000\text{ millimeters}

, a rate of

8\text{ millimeters per second}

is equal to

\frac{8}{1000} = 0.008\text{ meters per second}

.

To convert from a smaller unit (millimeters) to a larger unit (meters), divide by the conversion factor of

1000

.

2

Convert the time unit from seconds to hours.

Since

1\text{ hour} = 3600\text{ seconds}

, a rate of

0.008\text{ meters per second}

is equal to

0.008 \times 3600 = 28.8\text{ meters per hour}

.

To find the distance printed in one hour, multiply the distance printed in one second by the number of seconds in one hour (

3600

).

Key Concept

Unit Conversions

Alternative Method

We can combine the conversion steps into a single conversion factor:

\frac{3600\text{ seconds/hour}}{1000\text{ millimeters/meter}} = 3.6\text{ (meters \cdot seconds) / (millimeters \cdot hour)}

. Multiplying the rate of

8\text{ mm/s}

by

3.6

gives the rate directly as

28.8\text{ m/h}

.

Estimated Time:1m 0s

Question 5Question →

A bakery recipe requires $12$ ounces of butter to make one batch of bread. How many pounds of butter are needed to make $8$ batches of bread? (Given that $1\text{ pound} = 16\text{ ounces}$ )

Show answer & explanation

Answer: 6

Answer

The bakery needs

6

pounds of butter.

The total amount of butter needed for

8

batches is calculated by multiplying the butter required for one batch by the number of batches:

8 \times 12 = 96

ounces. Given that

1\text{ pound} = 16\text{ ounces}

, we convert ounces to pounds by dividing the total ounces by

16

. Therefore,

96 \div 16 = 6

pounds of butter are needed.

Step-by-Step Solution

1

Calculate the total ounces of butter needed.

96

ounces

Since each batch requires

12

ounces,

8

batches require

8 \times 12 = 96

ounces.

2

Convert the total ounces to pounds.

6

pounds

Divide the total number of ounces by the conversion factor (

16

ounces per pound):

\frac{96}{16} = 6

.

Key Concept

To convert a quantity from a smaller unit to a larger unit, multiply the initial quantity by the number of groups to find the total in the smaller unit, and then divide by the conversion factor that relates the two units.

Question 6Question →

An agricultural drone sprays liquid fertilizer at a rate of $1.5$ fluid ounces per square yard. The drone travels at a constant speed of $4.5$ miles per hour, spraying a continuous path that is exactly $6$ feet wide. Which of the following is closest to the rate at which the drone sprays the fertilizer, in gallons per minute? (Given that $1\text{ mile} = 5,280\text{ feet}$ , $1\text{ yard} = 3\text{ feet}$ , and $1\text{ gallon} = 128\text{ fluid ounces}$ )

3.09

3.58

9.28

27.84

Show answer & explanation

Answer: 3.09

Answer

The rate of

3.09

gallons per minute is closest to the drone's fertilizer spraying rate.

The correct answer is

3.09

because converting the speed of

4.5

miles per hour gives

396

feet per minute. Multiplying this by the width of

6

feet results in

2,376

square feet per minute. Dividing this by

9

converts the rate to

264

square yards per minute. Multiplying by the spray density of

1.5

fluid ounces per square yard gives

396

fluid ounces per minute. Finally, dividing by

128

fluid ounces per gallon yields approximately

3.09

gallons per minute.

Step-by-Step Solution

1

Convert the speed of the drone from miles per hour to feet per minute.

4.5\text{ miles/hour} = 396\text{ feet/minute}

To find the rate of area covered per minute, the speed must first be expressed in feet per minute. Multiply

4.5

by

5,280

to convert to feet per hour, then divide by

60

to convert to feet per minute:

\frac{4.5 \times 5,280}{60} = 396

.

2

Calculate the area covered by the drone per minute in square feet.

396\text{ feet/minute} \times 6\text{ feet} = 2,376\text{ square feet/minute}

Multiplying the travel speed in feet per minute by the path width of

6

feet gives the area covered per minute in square feet.

3

Convert the area rate from square feet per minute to square yards per minute.

2,376\text{ square feet/minute} \div 9 = 264\text{ square yards/minute}

Since

1\text{ yard} = 3\text{ feet}

, a square yard is

3 \times 3 = 9

square feet. Therefore, divide the area in square feet by

9

to convert to square yards.

4

Determine the fertilizer spray rate in fluid ounces per minute.

264\text{ square yards/minute} \times 1.5\text{ fluid ounces/square yard} = 396\text{ fluid ounces/minute}

Multiplying the area covered per minute by the spray density yields the volume of fertilizer sprayed per minute.

5

Convert the spray rate from fluid ounces per minute to gallons per minute.

396\text{ fluid ounces/minute} \div 128 \approx 3.09\text{ gallons/minute}

Since

1\text{ gallon} = 128\text{ fluid ounces}

, divide the fluid ounces per minute by

128

to find the rate in gallons per minute.

Key Concept

Unit conversions involving compound rates and area scaling.

Estimated Time:3m 0s

Question 7Question →

During a science experiment, a liquid is cooled at a constant rate of $0.5$ degrees Celsius per minute. What is the cooling rate of the liquid, in degrees Celsius per hour?

Show answer & explanation

Answer: 30

Answer

The cooling rate of the liquid is

30

degrees Celsius per hour.

To find the cooling rate in degrees Celsius per hour, we multiply the rate per minute by the conversion factor. There are

60

minutes in

1

hour. Multiplying

0.5

degrees Celsius per minute by

60

minutes per hour yields

30

degrees Celsius per hour.

Step-by-Step Solution

1

Identify the initial cooling rate per minute.

0.5

degrees Celsius per minute

This is the rate given in the problem statement.

2

Convert minutes to hours by multiplying the rate by

60

.

30

degrees Celsius per hour

An hour contains

60

minutes, so the total temperature decrease in one hour will be

60

times the decrease in one minute.

Key Concept

Unit conversions involving rates

Question 8Question →

A deep space satellite transmits data to a ground station at a constant rate of $1.6$ megabits per second. The transmitted data consists of actual scientific data and protocol overhead. The protocol overhead accounts for $25\%$ of the total transmitted bits, meaning the actual scientific data constitutes only $75\%$ of the total transmitted bits. The ground station needs to receive $5$ scientific data files, each with a size of $1.08$ gigabytes. How many hours will it take to transmit all $5$ files? (Given that $1\\text{ byte} = 8\\text{ bits}$ , $1\\text{ megabit} = 10^6\\text{ bits}$ , and $1\\text{ gigabyte} = 10^9\\text{ bytes}$ .)

Show answer & explanation

Answer: 10

Answer

It will take 10 hours to transmit all 5 files.

The correct answer is 10. To find this, first calculate the total size of the scientific files: 5 files * 1.08 gigabytes per file = 5.4 gigabytes. Convert this to bytes: 5.4 * 10^9 bytes. Convert bytes to bits: 5.4 * 10^9 * 8 = 43.2 * 10^9 bits. Since protocol overhead is 25%, the scientific data is 75% of the total transmitted bits, so the total bits transmitted is 43.2 * 10^9 / 0.75 = 57.6 * 10^9 bits. The transmission rate is 1.6 megabits per second, or 1.6 * 10^6 bits per second. The transmission time in seconds is 57.6 * 10^9 / (1.6 * 10^6) = 36,000 seconds. Converting seconds to hours: 36,000 / 3,600 = 10 hours.

Step-by-Step Solution

1

Calculate the total size of the scientific files in gigabytes.

5.4

gigabytes

To find the total amount of scientific data that must be received.

2

Convert the total scientific data size from gigabytes to bytes and then to bits.

43.2 \\times 10^9

bits

The transmission rate is given in megabits per second, so the data size must be converted to bits for unit consistency. Since

1\\text{ GB} = 10^9\\text{ bytes}

and

1\\text{ byte} = 8\\text{ bits}

, we have

5.4 \\times 10^9 \\times 8 = 43.2 \\times 10^9

bits.

3

Calculate the total number of bits transmitted, including protocol overhead.

57.6 \\times 10^9

bits

Since protocol overhead is

25\%

, the scientific data is only

75\%

of the total bits transmitted. Thus, we divide the scientific bits by

0.75

to find the total bits transmitted:

\\frac{43.2 \\times 10^9}{0.75} = 57.6 \\times 10^9

bits.

4

Calculate the transmission time in seconds.

36,000

seconds

Divide the total bits by the transmission rate in bits per second (

1.6\\text{ Mbps} = 1.6 \\times 10^6

bits per second):

\\frac{57.6 \\times 10^9}{1.6 \\times 10^6} = 36,000

seconds.

5

Convert the transmission time from seconds to hours.

10

hours

Since there are

3,600

seconds in one hour, divide the total seconds by

3,600

:

\\frac{36,000}{3,600} = 10

.

Key Concept

Multi-step dimensional analysis and compound rate conversions incorporating percentage overhead

Alternative Method

Instead of converting units step-by-step, dimensional analysis can be set up as a single product of conversion factors: 5 files * (1.08 GB / 1 file) * (10^9 bytes / 1 GB) * (8 bits / 1 byte) * (1 total bit / 0.75 scientific bits) * (1 second / 1.6 * 10^6 bits) * (1 hour / 3600 seconds) = 10 hours.

Estimated Time:3m 0s

Question 9Question →

An industrial printing press uses ink at a constant rate of $0.05$ milliliters per square centimeter of printed paper. The press prints a continuous roll of paper that is $40$ centimeters wide at a constant speed of $2$ meters per second. At this rate, how many liters of ink does the press use during $1$ hour of continuous printing?

Show answer & explanation

Answer: 1440

Answer

1440

The correct answer is 1440. First, convert the speed of the paper from meters per second to centimeters per second:

2 \text{ m/s} \times 100 \text{ cm/m} = 200 \text{ cm/s}

. Next, find the rate at which the paper surface area is printed:

40 \text{ cm} \times 200 \text{ cm/s} = 8,000 \text{ cm}^2/\text{s}

. Multiply this by the ink usage rate to find milliliters of ink per second:

8,000 \text{ cm}^2/\text{s} \times 0.05 \text{ mL/cm}^2 = 400 \text{ mL/s}

. Since there are

3,600

seconds in

1

hour, the press uses

400 \text{ mL/s} \times 3,600 \text{ s} = 1,440,000 \text{ mL}

of ink per hour. Finally, convert milliliters to liters:

1,440,000 \text{ mL} \div 1,000 \text{ mL/L} = 1,440

liters.

Step-by-Step Solution

1

Convert the speed of the paper from meters per second to centimeters per second.

200 \text{ cm/s}

To align the speed unit with the paper width unit (

40 \text{ cm}

).

2

Calculate the surface area of paper printed per second.

8,000 \text{ cm}^2/\text{s}

Multiply the width of the paper (

40 \text{ cm}

) by the converted speed (

200 \text{ cm/s}

).

3

Calculate the volume of ink used per second in milliliters.

400 \text{ mL/s}

Multiply the area printed per second (

8,000 \text{ cm}^2/\text{s}

) by the ink consumption rate (

0.05 \text{ mL/cm}^2

).

4

Convert the ink volume rate from per second to per hour.

1,440,000 \text{ mL/h}

Multiply the rate per second by

3,600

, since there are

3,600

seconds in

1

hour.

5

Convert the final volume from milliliters to liters.

1440

Divide the total milliliters (

1,440,000

) by

1,000

, since

1

liter is equal to

1,000

milliliters.

Key Concept

Multi-step dimensional analysis and compound unit conversion (length, area, volume, and time).

Question 10Question →

A solar power station consists of solar panels that generate electricity at a constant rate of $15\text{ watts}$ per square foot of panel surface area (where $1\text{ watt} = 1\text{ joule per second}$ ). The panels have a total surface area of $12,000\text{ square yards}$ . How many megajoules (MJ) of energy do the panels generate in $2\text{ hours}$ ? (Given that $1\text{ yard} = 3\text{ feet}$ and $1\text{ MJ} = 1,000,000\text{ joules}$ )

144

194.4

3,888

11,664

Show answer & explanation

Answer: $11,664$

Answer

The solar panels generate

11,664

megajoules of energy in

2\text{ hours}

.

To find the energy generated, first convert the area from square yards to square feet. Since

1\text{ yard} = 3\text{ feet}

,

1\text{ square yard} = (3\text{ feet})^2 = 9\text{ square feet}

. Multiplying

12,000\text{ square yards}

by

9

gives

108,000\text{ square feet}

. The power generated is

108,000\text{ square feet} \times 15\text{ watts/square foot} = 1,620,000\text{ watts}

, which is

1,620,000\text{ joules per second}

. In

2\text{ hours}

, which is

2 \times 3,600 = 7,200\text{ seconds}

, the total energy generated is

1,620,000\text{ joules/second} \times 7,200\text{ seconds} = 11,664,000,000\text{ joules}

. Since

1\text{ MJ} = 1,000,000\text{ joules}

, this is equivalent to

11,664\text{ MJ}

.

Step-by-Step Solution

1

Convert the surface area of the solar panels from square yards to square feet.

Since

1\text{ yard} = 3\text{ feet}

,

1\text{ square yard} = 3^2 = 9\text{ square feet}

. Therefore, the total area in square feet is

12,000 \times 9 = 108,000\text{ square feet}

.

To apply the generation rate given in watts per square foot, the area must be in square feet.

2

Determine the total rate of energy generation in joules per second.

The total power generated is

108,000\text{ square feet} \times 15\text{ watts per square foot} = 1,620,000\text{ watts}

. Since

1\text{ watt} = 1\text{ joule per second}

, this equals

1,620,000\text{ joules per second}

.

This calculates the total energy produced by the entire solar farm every second.

3

Convert the total time from hours to seconds.

2\text{ hours} \times 60\text{ minutes/hour} \times 60\text{ seconds/minute} = 7,200\text{ seconds}

.

To find the total energy generated, the rate (joules per second) must be multiplied by the duration in seconds.

4

Calculate the total energy generated in joules, then convert it to megajoules (MJ).

Total energy in joules is

1,620,000\text{ joules/second} \times 7,200\text{ seconds} = 11,664,000,000\text{ joules}

. Converting to megajoules:

11,664,000,000\text{ joules} \div 1,000,000\text{ joules/MJ} = 11,664\text{ MJ}

.

This yields the final quantity of energy in the requested units.

Key Concept

Multi-step unit conversions involving area scaling, compound rates, and time conversions.

Estimated Time:2m 30s

Question 11Question →

A manufacturing plant uses a machine lubricant at a constant rate of $0.08$ fluid ounces per second of operation. The plant operates the machinery for $8$ hours per day, $5$ days a week. The lubricant is purchased in drums, where each drum contains $15$ gallons of lubricant. How many drums of lubricant does the plant use in a period of $5$ weeks? (Note: $1\text{ gallon} = 128\text{ fluid ounces}$ )

Show answer & explanation

Answer: 30

Answer

The plant uses 30 drums of lubricant in 5 weeks.

To find the number of drums, calculate the total seconds of operation: 5 weeks * 5 days/week * 8 hours/day * 3,600 seconds/hour = 720,000 seconds. Multiply this by the consumption rate of 0.08 fluid ounces/second to get 57,600 fluid ounces. Convert this to gallons by dividing by 128 fluid ounces/gallon to get 450 gallons. Finally, divide by 15 gallons/drum to find that 30 drums are used.

Step-by-Step Solution

1

Calculate the total number of operational seconds in the 5-week period.

720{}000\text{ seconds}

Multiply 5 weeks by 5 days per week, 8 hours per day, and 3,600 seconds per hour:

5 \times 5 \times 8 \times 3{}600 = 720{}000

.

2

Calculate the total volume of lubricant used in fluid ounces.

57{}600\text{ fluid ounces}

Multiply the rate of

0.08\text{ fluid ounces per second}

by the total operational seconds:

720{}000 \times 0.08 = 57{}600

.

3

Convert the total fluid ounces to gallons.

450\text{ gallons}

Divide the total fluid ounces by the conversion factor of

128\text{ fluid ounces per gallon}

:

57{}600 / 128 = 450

.

4

Calculate the total number of drums needed.

30\text{ drums}

Divide the total gallons by the capacity of a single drum (

15\text{ gallons}

):

450 / 15 = 30

.

Key Concept

Multi-step dimensional analysis and unit conversion under compound rates

Question 12Question →

A cargo ship travels at a constant speed of $24\text{ knots}$ . The ship's engine consumes fuel at a constant rate of $23\text{ liters per minute}$ . The fuel has a density of $0.80\text{ grams per milliliter}$ . Given that $1\text{ knot} = 1.15\text{ miles per hour}$ and $1\text{ metric ton} = 1,000\text{ kilograms}$ , what is the ship's fuel consumption rate in metric tons per mile?

0.04

0.053

0.0625

0.40

Show answer & explanation

Answer: 0.04

Answer

0.04

The correct answer is obtained by first converting the ship's speed to miles per hour (

24 \times 1.15 = 27.6\text{ mph}

). Next, the fuel usage is converted to liters per hour (

23 \times 60 = 1,380\text{ L/hr}

). Using the density conversion where

0.80\text{ g/mL}

is equal to

0.80\text{ kg/L}

, the fuel consumption is converted to mass (

1,380 \times 0.80 = 1,104\text{ kg/hr}

), which is equivalent to

1.104

metric tons per hour. Finally, dividing the fuel rate by the speed gives the rate per mile:

1.104 / 27.6 = 0.04

metric tons per mile.

Step-by-Step Solution

1

Convert the speed of the cargo ship from knots to miles per hour.

24\text{ knots} \times 1.15\text{ miles per hour per knot} = 27.6\text{ miles per hour}

To match the final unit of miles, we convert knots to miles per hour.

2

Convert the fuel consumption rate from liters per minute to liters per hour.

23\text{ liters/minute} \times 60\text{ minutes/hour} = 1,380\text{ liters/hour}

Since speed is in miles per hour, we convert the fuel consumption to a per-hour rate.

3

Convert the volume consumption rate to a mass consumption rate using the density.

1,380\text{ liters/hour} \times 0.80\text{ kilograms/liter} = 1,104\text{ kilograms/hour}

Density of

0.80\text{ g/mL}

is equivalent to

0.80\text{ kg/L}

because there are

1,000

grams in a kilogram and

1,000

milliliters in a liter.

4

Convert the mass rate from kilograms per hour to metric tons per hour.

\frac{1,104\text{ kilograms/hour}}{1,000\text{ kilograms/metric ton}} = 1.104\text{ metric tons/hour}

To match the final unit of metric tons, we divide the kilograms by

1,000

.

5

Calculate the fuel consumption rate in metric tons per mile by dividing the hourly fuel consumption by the hourly speed.

\frac{1.104\text{ metric tons/hour}}{27.6\text{ miles/hour}} = 0.04\text{ metric tons/mile}

Dividing the rate of fuel consumed per hour by the distance traveled per hour gives the fuel consumed per mile.

Key Concept

Unit conversions with compound rates and density scaling

Estimated Time:2m 30s

Question 13Question →

An industrial water filtration system filters water at a constant rate of $0.5\text{ liters per second}$ . The system consumes a chemical powder at a rate of $2.5\text{ milligrams}$ per gram of impurities removed from the water. If the water contains an average of $0.12\text{ grams}$ of impurities per liter, how many grams of the chemical powder does the system consume during $4\text{ hours}$ of continuous operation?

Show answer & explanation

Answer: 2.16

Answer

The system consumes

2.16\text{ grams}

of chemical powder during the

4

hours of operation.

The correct answer is

2.16

. First, convert the

4

hours of continuous operation into seconds:

4 \times 3,600 = 14,400

seconds. Next, multiply this duration by the filtration rate to find the total water filtered:

14,400 \times 0.5 = 7,200

liters. Multiply the volume by the impurity concentration to find the total mass of impurities:

7,200 \times 0.12 = 864

grams. Then, find the mass of chemical powder consumed:

864 \times 2.5 = 2,160

milligrams. Finally, convert milligrams to grams by dividing by

1,000

, resulting in

2.16

grams.

Step-by-Step Solution

1

Convert the operating time from hours to seconds.

14,400\text{ seconds}

Since the filtration rate is given in liters per second, the total operating time must be converted from hours to seconds to align units. There are

3,600

seconds in one hour.

2

Calculate the total volume of water filtered during the operation.

7,200\text{ liters}

Multiply the operating time in seconds by the filtration rate of

0.5

liters per second.

3

Calculate the total mass of impurities removed from the filtered water.

864\text{ grams}

Multiply the total volume of filtered water by the concentration rate of

0.12

grams of impurities per liter.

4

Determine the mass of chemical powder consumed in milligrams.

2,160\text{ milligrams}

Multiply the total mass of impurities in grams by the consumption rate of

2.5

milligrams of chemical powder per gram of impurities.

5

Convert the mass of chemical powder from milligrams to grams.

2.16\text{ grams}

Divide the mass in milligrams by

1,000

because there are

1,000

milligrams in one gram.

Key Concept

Dimensional analysis and multi-step unit conversion involving rates

Question 14Question →

A 3D printer uses a plastic filament at a constant rate of $2.5\text{ grams}$ per minute of printing time. The filament has a density of $1.25\text{ grams per cubic centimeter}$ . If the printer runs continuously for $4\text{ hours}$ , what is the total volume, in cubic centimeters, of the filament used?

8

480

600

750

Show answer & explanation

Answer: 480

Answer

480

The correct answer is 480. First, convert the printing time from hours to minutes: 4 hours multiplied by 60 minutes per hour equals 240 minutes. Next, calculate the total mass of the filament used by multiplying the printing time by the usage rate: 240 minutes multiplied by 2.5 grams per minute equals 600 grams. Finally, convert the mass to volume by dividing the mass by the density: 600 grams divided by 1.25 grams per cubic centimeter equals 480 cubic centimeters.

Step-by-Step Solution

1

Convert the printing time from hours to minutes.

240\text{ minutes}

Since the filament consumption rate is given in grams per minute, the total time must be in minutes. Multiplying

4\text{ hours}

by

60\text{ minutes per hour}

yields

240\text{ minutes}

.

2

Calculate the total mass of the filament used.

600\text{ grams}

Multiplying the total printing time of

240\text{ minutes}

by the consumption rate of

2.5\text{ grams per minute}

gives a total mass of

600\text{ grams}

.

3

Calculate the total volume of the filament used.

480\text{ cubic centimeters}

Divide the total mass of

600\text{ grams}

by the density of

1.25\text{ grams per cubic centimeter}

to find the volume:

\frac{600}{1.25} = 480\text{ cubic centimeters}

.

Key Concept

Multi-unit dimensional analysis involving rates, time, and density.

Question 15Question →

An electric vehicle consumes $0.35$ kilowatt-hours ( $\text{kWh}$ ) of electricity per mile traveled. The vehicle travels at a constant speed of $48$ miles per hour. What is the vehicle's rate of electricity consumption, in $\text{kWh}$ per minute?

Show answer & explanation

Answer: 0.28

Answer

0.28

To find the rate of electricity consumption in kilowatt-hours (

\text{kWh}

) per minute, we must convert the given rate of

0.35\text{ kWh}

per mile into

\text{kWh}

per minute. First, convert the speed of

48\text{ miles per hour}

to miles per minute:

\frac{48\text{ miles}}{60\text{ minutes}} = 0.8\text{ miles per minute}

. Next, multiply the energy consumed per mile by the distance traveled per minute:

0.35\text{ kWh per mile} \times 0.8\text{ miles per minute} = 0.28\text{ kWh per minute}

. Therefore, the vehicle's rate of electricity consumption is

0.28\text{ kWh}

per minute.

Step-by-Step Solution

1

Convert the vehicle's speed from miles per hour to miles per minute.

0.8

miles per minute

Since there are 60 minutes in 1 hour, dividing the speed in miles per hour by 60 gives the speed in miles per minute:

\frac{48\text{ miles}}{1\text{ hour}} \times \frac{1\text{ hour}}{60\text{ minutes}} = 0.8\text{ miles per minute}

.

2

Calculate the rate of electricity consumption in kWh per minute.

0.28

kWh per minute

Multiply the consumption rate per mile by the distance traveled per minute:

0.35\text{ kWh per mile} \times 0.8\text{ miles per minute} = 0.28\text{ kWh per minute}

.

Key Concept

Unit conversion involving rates and compound units.

Alternative Method

First, find the total electricity consumed in one hour of driving by multiplying the consumption rate per mile by the miles traveled in one hour:

0.35\text{ kWh per mile} \times 48\text{ miles} = 16.8\text{ kWh}

. Then, convert this hourly rate to a minute rate by dividing by 60 minutes:

\frac{16.8\text{ kWh}}{60\text{ minutes}} = 0.28\text{ kWh per minute}

.

Estimated Time:1m 15s

Question 16Question →

An aquarium filter recirculates water at a constant rate of $250$ milliliters per second. The filter cartridge must be replaced after it has processed a total of $540$ cubic meters of water. For how many hours of continuous operation can the filter run before the cartridge must be replaced? ( $1\text{ cubic meter} = 1,000\text{ liters}$ )

Show answer & explanation

Answer: 600

Answer

The filter can run for

600

hours of continuous operation before the cartridge must be replaced.

To find the number of hours the filter can run, we first convert the flow rate to cubic meters per hour. The flow rate of

250

milliliters per second is equal to

0.25

liters per second. Multiplying by

3,600

seconds per hour gives a flow rate of

900

liters per hour. Since

1\text{ cubic meter} = 1,000\text{ liters}

,

900

liters per hour is equivalent to

0.9

cubic meters per hour. Finally, dividing the total capacity of

540

cubic meters by the flow rate of

0.9

cubic meters per hour yields a duration of

600

hours.

Step-by-Step Solution

1

Convert the water recirculating rate from milliliters per second to liters per second.

0.25

liters per second

To express the flow rate in terms of liters before converting to cubic meters.

2

Convert the rate from liters per second to liters per hour.

900

liters per hour

To align the time unit of the rate with the requested time unit (hours).

3

Convert the rate from liters per hour to cubic meters per hour.

0.9

cubic meters per hour

To match the volume unit of the filter cartridge capacity (cubic meters).

4

Divide the total cartridge capacity by the hourly flow rate of the filter.

600

hours

To find the total duration of continuous operation before replacement.

Key Concept

Unit Conversions

Question 17Question →

A wind turbine generates electricity at a constant rate of $1.5\text{ megawatt-hours (MWh)}$ per day. If $1\text{ megawatt-hour}$ is equivalent to $3.6\text{ gigajoules (GJ)}$ , how many gigajoules of energy does the wind turbine generate in $8\text{ hours}$ ?

16.2

1.8

43.2

0.14

Show answer & explanation

Answer: 1.8

Answer

1.8

The correct answer is 1.8. To find the total energy generated in gigajoules over 8 hours, the daily rate of

1.5\text{ MWh}

is first converted to an hourly rate by dividing by 24 hours, yielding

0.0625\text{ MWh}

per hour. Multiplying this rate by 8 hours results in

0.5\text{ MWh}

of energy generated. Finally, multiplying the energy in MWh by the conversion factor of

3.6\text{ GJ}

per MWh yields

1.8\text{ GJ}

.

Step-by-Step Solution

1

Convert the daily generation rate of

1.5\text{ MWh}

to an hourly generation rate by dividing by 24 hours.

0.0625\text{ MWh per hour}

Since there are 24 hours in a day, dividing the daily rate by 24 yields the rate of energy generated per hour.

2

Multiply the hourly generation rate by 8 hours to find the total energy generated in megawatt-hours.

0.5\text{ MWh}

Multiplying the rate per hour by the duration of 8 hours gives the cumulative energy generated over that period.

3

Convert the energy generated from megawatt-hours to gigajoules by multiplying by the conversion factor of

3.6\text{ GJ}

per MWh.

1.8\text{ GJ}

Multiplying the energy in MWh by

3.6\text{ GJ/MWh}

converts the unit to gigajoules.

Key Concept

Unit Conversions

Question 18Question →

An agricultural drone sprays a liquid fertilizer at a constant rate of $120$ milliliters per second. The drone flies at a constant speed, covering $1.5$ hectares per hour. If the drone operates continuously at this rate, how many liters of fertilizer are sprayed per hectare?

(Note: $1$ liter = $1,000$ milliliters)

4.8

288

432

648

Show answer & explanation

Answer: 288

Answer

The drone sprays 288 liters of fertilizer per hectare.

To find the number of liters sprayed per hectare, the rate of spraying must be converted from milliliters per second to liters per hour, and then divided by the coverage rate of hectares per hour. First, multiply the spray rate of

120

milliliters per second by the number of seconds in an hour (

3,600

seconds) to find that the drone sprays

432,000

milliliters per hour. Next, convert this volume to liters by dividing by

1,000

milliliters per liter, which gives

432

liters per hour. Finally, divide this hourly spray volume by the coverage rate of

1.5

hectares per hour to obtain

288

liters per hectare.

Step-by-Step Solution

1

Calculate the volume of fertilizer sprayed in one hour in milliliters.

120\text{ mL/s} \times 3,600\text{ s/h} = 432,000\text{ mL/h}

There are

3,600

seconds in one hour, so multiplying the per-second rate by

3,600

gives the volume sprayed per hour.

2

Convert the hourly volume from milliliters to liters.

\frac{432,000\text{ mL/h}}{1,000\text{ mL/L}} = 432\text{ L/h}

Since

1

liter is equivalent to

1,000

milliliters, divide the volume in milliliters by

1,000

to get the volume in liters.

3

Calculate the volume of fertilizer sprayed per hectare.

\frac{432\text{ L/h}}{1.5\text{ hectares/h}} = 288\text{ L/hectare}

Divide the total liters sprayed in one hour by the total hectares covered in that same hour to find the rate per hectare.

Key Concept

Unit conversion involving compound rates of volume, time, and area.

Question 19Question →

A paper mill machine produces paper at a constant rate of $24$ square feet per second. If $1\text{ square yard} = 9\text{ square feet}$ , how many square yards of paper does the machine produce in $15$ minutes?

40

2,400

21,600

194,400

Show answer & explanation

Answer: 2,400

Answer

2,400

The correct answer of 2,400 is found by first converting the time of 15 minutes to seconds:

15 \times 60 = 900

seconds. Next, calculate the total area produced in square feet by multiplying the production rate by the time:

24 \times 900 = 21,600

square feet. Finally, convert this area to square yards by dividing by 9:

21,600 / 9 = 2,400

square yards.

Step-by-Step Solution

1

Convert the time from minutes to seconds.

15\text{ minutes} \times 60\text{ seconds/minute} = 900\text{ seconds}

The production rate is given in square feet per second, so the time duration must be in seconds.

2

Calculate the total area of paper produced in square feet.

24\text{ square feet/second} \times 900\text{ seconds} = 21,600\text{ square feet}

Multiplying the rate by the total time gives the total amount produced in square feet.

3

Convert the total area from square feet to square yards.

21,600\text{ square feet} \div 9\text{ square feet/square yard} = 2,400\text{ square yards}

Since

1\text{ square yard} = 9\text{ square feet}

, dividing the total square feet by 9 converts the value to square yards.

Key Concept

Multi-step unit conversions involving rates of change and area scaling.

Question 20Question →

A specialized laser cutter cuts steel sheet metal at a constant rate of $18$ inches per minute. At this rate, how many seconds does it take the laser cutter to cut a length of $6$ feet of steel? ( $1\text{ foot} = 12\text{ inches}$ )

Show answer & explanation

Answer: 240

Answer

The correct answer is 240.

To find the number of seconds it takes the laser cutter to cut

6

feet of steel, we first convert the length to inches:

6\text{ feet} \times 12\text{ inches/foot} = 72\text{ inches}

. Next, we divide this length by the cutting rate to find the duration in minutes:

72\text{ inches} \div 18\text{ inches/minute} = 4\text{ minutes}

. Finally, we convert the minutes into seconds: $4\text{ minutes} \times 60\text{ seconds/minute} = 240\text{ seconds}.

Step-by-Step Solution

1

Convert the total cutting distance from feet to inches.

72\text{ inches}

Since the rate is given in inches per minute, we must convert the length from feet to inches to align the units. Knowing that

1\text{ foot} = 12\text{ inches}

, we multiply

6

by

12

.

2

Calculate the time in minutes to cut

72\text{ inches}

at a rate of

18\text{ inches/minute}

.

4\text{ minutes}

Dividing the total distance (

72\text{ inches}

) by the cutting rate (

18\text{ inches/minute}

) yields the time in minutes.

3

Convert the time from minutes to seconds.

240\text{ seconds}

Since

1\text{ minute} = 60\text{ seconds}

, multiplying

4

minutes by

60

gives the final time in seconds.

Key Concept

Performing multi-step unit conversions using rate and dimensional analysis.

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