Solved Exercises on Dimensional Analysis in Integrated Math 1

Master dimensional analysis: unit conversion, rate problems, complex conversions, and multi-step unit problems through these 5 detailed exercises with comprehensive solutions.

Solution: Exercises 1 to 3
1 Simple unit conversion
Exercise 1
Convert 2.5 feet to inches. (1 foot = 12 inches)
Definition:

Dimensional analysis: A mathematical method that uses conversion factors to change units while preserving the actual quantity

Conversion factor: A ratio equal to 1 that relates two different units

Unit cancellation: The process of eliminating unwanted units by multiplication and division

Dimensional analysis method:

To convert units using dimensional analysis:

  1. Identify the starting and ending units
  2. Find the conversion factor
  3. Set up the conversion factor as a fraction
  4. Arrange so the starting unit cancels out
  5. Multiply and divide to get the answer
  6. Verify the final unit is correct
Given
\(2.5 \text{ feet}\)
Conversion factor
\(\frac{12 \text{ inches}}{1 \text{ foot}}\)
Multiply
\(2.5 \times 12 = 30\)
Final unit
inches
Step 1: Write the starting quantity

2.5 feet

Step 2: Identify the conversion factor

Given: 1 foot = 12 inches

So: \(\frac{12 \text{ inches}}{1 \text{ foot}} = 1\)

Step 3: Set up the multiplication

Place the conversion factor so that feet cancel:

\(2.5 \text{ feet} \times \frac{12 \text{ inches}}{1 \text{ foot}}\)

Step 4: Cancel units

The "feet" in the numerator and denominator cancel out:

\(2.5 \cancel{\text{feet}} \times \frac{12 \text{ inches}}{1 \cancel{\text{foot}}}\)

Step 5: Calculate the numerical result

\(2.5 \times 12 = 30\)

Step 6: Express the final answer

Result: 30 inches

30 inches
Final answer:

2.5 feet equals 30 inches

Applied rules:

• Unit Conversion: Multiply by a conversion factor equal to 1

• Unit Cancellation: Place units strategically to cancel unwanted units

• Conversion Factor: Express the same quantity in different units

Tip: Always arrange conversion factors so the unit you want to cancel is in both numerator and denominator!
Tip: Write units with your numbers to track cancellations!
2 Rate conversion
Exercise 2
Convert 60 miles per hour to feet per second. (1 mile = 5280 feet, 1 hour = 3600 seconds)
Definition:

Rate conversion: Converting a rate from one unit of measurement to another

Composite conversion: Converting both distance and time units simultaneously

Unit analysis: Tracking units through calculations to ensure correctness

Start
\(60 \frac{\text{miles}}{\text{hour}}\)
Convert miles to feet
\(60 \times \frac{5280}{1} = 316800 \frac{\text{feet}}{\text{hour}}\)
Convert hours to seconds
\(316800 \times \frac{1}{3600} = 88 \frac{\text{feet}}{\text{second}}\)
Step 1: Write the starting rate

\(60 \frac{\text{miles}}{\text{hour}}\)

Step 2: Set up conversion factors

We need: miles → feet and hours → seconds

Given: 1 mile = 5280 feet, 1 hour = 3600 seconds

Step 3: Arrange conversion factors

For miles to feet: \(\frac{5280 \text{ feet}}{1 \text{ mile}}\) (miles cancel out)

For hours to seconds: \(\frac{1 \text{ hour}}{3600 \text{ seconds}}\) (hours cancel out)

Step 4: Multiply the conversion factors

\(60 \frac{\text{miles}}{\text{hour}} \times \frac{5280 \text{ feet}}{1 \text{ mile}} \times \frac{1 \text{ hour}}{3600 \text{ seconds}}\)

Step 5: Cancel units and calculate

\(60 \times \frac{5280}{3600} = \frac{316800}{3600} = 88\)

Step 6: Write the final rate

\(88 \frac{\text{feet}}{\text{second}}\)

\(88 \frac{\text{feet}}{\text{second}}\)
Final answer:

60 miles per hour equals 88 feet per second

Applied rules:

• Rate Conversion: Convert one unit at a time while preserving the rate relationship

• Unit Cancellation: Arrange conversion factors so unwanted units cancel

• Sequential Conversion: Handle distance and time conversions separately

Tip: When converting rates, handle distance and time units separately!
Tip: Always verify that your final units match what the question asks for!
3 Volume conversion
Exercise 3
A container holds 3.5 gallons of liquid. Convert this to cubic inches. (1 gallon = 231 cubic inches)
Definition:

Volume conversion: Converting between different units of volume

Direct conversion: Using a single conversion factor to convert between units

Volume relationships: Understanding how different volume units relate to each other

Given
\(3.5 \text{ gallons}\)
Conversion factor
\(\frac{231 \text{ in}^3}{1 \text{ gallon}}\)
Multiply
\(3.5 \times 231 = 808.5\)
Final unit
cubic inches
Step 1: Identify the given quantity

3.5 gallons

Step 2: Find the conversion factor

Given: 1 gallon = 231 cubic inches

So: \(\frac{231 \text{ in}^3}{1 \text{ gallon}} = 1\)

Step 3: Set up the multiplication

\(3.5 \text{ gallons} \times \frac{231 \text{ in}^3}{1 \text{ gallon}}\)

Step 4: Cancel units

The "gallons" in the numerator and denominator cancel out

\(3.5 \cancel{\text{gallons}} \times \frac{231 \text{ in}^3}{1 \cancel{\text{gallon}}}\)

Step 5: Calculate the result

\(3.5 \times 231 = 808.5\)

Step 6: Express the final answer

Result: 808.5 cubic inches

808.5 in³
Final answer:

The container holds 808.5 cubic inches of liquid

Applied rules:

• Volume Conversion: Use the appropriate volume conversion factor

• Unit Cancellation: Arrange factors so unwanted units cancel out

• Direct Proportion: When units are directly related, use single conversion factor

Tip: Common volume conversions: 1 gallon = 231 cubic inches = 3.785 liters.
Tip: Always cube the conversion factor when converting cubic units!
Rules and methods, laws,...
\(\text{Quantity}_1 \times \frac{\text{Unit}_2}{\text{Unit}_1} = \text{Quantity}_2\)
Dimensional Analysis Formula
\(\text{Rate}_1 \times \frac{\text{Unit}_{2a}}{\text{Unit}_{1a}} \times \frac{\text{Unit}_{1b}}{\text{Unit}_{2b}} = \text{Rate}_2\)
Rate Conversion Formula
Length
1 mile = 5280 ft
1 foot = 12 inches
Time
1 hour = 60 min
1 minute = 60 sec
Volume
1 gal = 231 in³
1 ft³ = 1728 in³
Weight
1 lb = 16 oz
1 ton = 2000 lbs
Unit Consistency: Units must be compatible in calculations.
Dimensional Analysis: Units must make sense in the final answer.
Solution: Exercises 4 to 5
4 Complex rate conversion
Exercise 4
A pump fills a tank at a rate of 8 gallons per minute. Convert this rate to cubic feet per hour. (1 gallon = 0.1337 cubic feet, 1 hour = 60 minutes)
Definition:

Complex rate conversion: Converting rates that involve multiple unit conversions simultaneously

Sequential conversion: Handling volume and time unit conversions in sequence

Rate equivalency: Ensuring the same amount of work is represented in different units

Start
\(8 \frac{\text{gallons}}{\text{minute}}\)
Convert gallons to cubic feet
\(8 \times 0.1337 = 1.0696 \frac{\text{ft}^3}{\text{minute}}\)
Convert minutes to hours
\(1.0696 \times 60 = 64.176 \frac{\text{ft}^3}{\text{hour}}\)
Step 1: Write the starting rate

\(8 \frac{\text{gallons}}{\text{minute}}\)

Step 2: Convert gallons to cubic feet

Using conversion factor: 1 gallon = 0.1337 cubic feet

\(8 \frac{\text{gallons}}{\text{minute}} \times \frac{0.1337 \text{ ft}^3}{1 \text{ gallon}} = 1.0696 \frac{\text{ft}^3}{\text{minute}}\)

Step 3: Convert minutes to hours

Using conversion factor: 1 hour = 60 minutes

\(1.0696 \frac{\text{ft}^3}{\text{minute}} \times \frac{60 \text{ minutes}}{1 \text{ hour}} = 64.176 \frac{\text{ft}^3}{\text{hour}}\)

Step 4: Verify units cancel correctly

Gallons cancel, minutes cancel, leaving cubic feet per hour

Step 5: Express the final answer

The pump rate is approximately 64.18 cubic feet per hour

\(64.18 \frac{\text{ft}^3}{\text{hour}}\)
Final answer:

The pump fills at a rate of approximately 64.18 cubic feet per hour

Applied rules:

• Sequential Conversion: Handle one unit conversion at a time

• Rate Preservation: The actual rate remains the same, only units change

• Unit Cancellation: Verify that all unwanted units cancel out

Tip: For complex conversions, convert one unit at a time!
5 Advanced dimensional analysis
Exercise 5
A car travels 45 miles per hour. Convert this speed to meters per second. (1 mile = 1609.34 meters, 1 hour = 3600 seconds)
Definition:

Advanced dimensional analysis: Converting between different measurement systems (imperial to metric)

International conversion: Using precise conversion factors between systems

Complex unit manipulation: Handling multiple conversions with decimal factors

Start
\(45 \frac{\text{miles}}{\text{hour}}\)
Convert miles to meters
\(45 \times 1609.34 = 72420.3 \frac{\text{meters}}{\text{hour}}\)
Convert hours to seconds
\(\frac{72420.3}{3600} = 20.12 \frac{\text{meters}}{\text{second}}\)
Step 1: Write the starting rate

\(45 \frac{\text{miles}}{\text{hour}}\)

Step 2: Convert miles to meters

Using conversion factor: 1 mile = 1609.34 meters

\(45 \frac{\text{miles}}{\text{hour}} \times \frac{1609.34 \text{ meters}}{1 \text{ mile}} = 72420.3 \frac{\text{meters}}{\text{hour}}\)

Step 3: Convert hours to seconds

Using conversion factor: 1 hour = 3600 seconds

\(72420.3 \frac{\text{meters}}{\text{hour}} \times \frac{1 \text{ hour}}{3600 \text{ seconds}} = 20.12 \frac{\text{meters}}{\text{second}}\)

Step 4: Verify the units

Miles cancel, hours cancel, leaving meters per second

Step 5: Express the final answer

The car travels approximately 20.12 meters per second

\(20.12 \frac{\text{meters}}{\text{second}}\)
Final answer:

The car travels at approximately 20.12 meters per second

Applied rules:

• System Conversion: Converting between imperial and metric systems

• Precision: Using accurate conversion factors for scientific calculations

• Decimal Arithmetic: Handling calculations with decimal conversion factors

Tip: For metric conversions, use 1609.34 meters per mile for precision!
Tip: Remember: 1 hour = 3600 seconds, not 60!
Comprehensive Guide to Dimensional Analysis
\(\text{Result} = \text{Initial Value} \times \prod \left(\frac{\text{New Unit}}{\text{Old Unit}}\right)\)
General Conversion Formula
Key definitions:

Dimensional analysis: Using units to guide calculations and verify results

Conversion factor: A ratio equal to 1 that converts between units

Unit consistency: Ensuring all units in a calculation are compatible

Unit cancellation: The process of eliminating unwanted units through division

Rate: A comparison of two quantities with different units

Complete methodology:
  1. Identify the problem: Determine what you're looking for and what you have
  2. List known conversions: Write down all relevant conversion factors
  3. Plan the path: Determine the sequence of conversions needed
  4. Set up conversions: Arrange conversion factors so units cancel appropriately
  5. Calculate: Perform the multiplication and division
  6. Verify: Check that units are correct and magnitude makes sense
Tip 1: Always write units with your numbers to track cancellations!
Tip 2: Arrange conversion factors so the unit you want to cancel is in both numerator and denominator!
Tip 3: Common time conversions: 60 sec/min, 60 min/hr, 24 hr/day.
Tip 4: Common length conversions: 12 in/ft, 3 ft/yd, 5280 ft/mile.
Tip 5: If your final units don't match what you expect, you made a setup error!
Common errors: Forgetting to write units, misplacing conversion factors, incorrect arithmetic, not cancelling units properly.
Exam preparation: Practice common conversion factors, focus on dimensional analysis, master unit cancellation.
Essential conversion facts:

• Length: 1 mile = 5280 feet = 1760 yards

• Time: 1 hour = 60 minutes = 3600 seconds

• Volume: 1 gallon = 3.785 liters = 231 cubic inches

• Weight: 1 pound = 16 ounces, 1 ton = 2000 pounds

• Metric: 1 meter = 100 centimeters = 1000 millimeters

Dimensional Analysis Workflow

📊
Problem-Solving Process
1
Identify Units
2
Find Conversion Factors
3
Arrange Factors
4
Multiply
5
Verify Units
Common Conversions
Length: mi → ft → in
Time: hr → min → sec
Volume: gal → L → mL
Weight: lb → oz → g
Distance: km → mi → m
Speed: mi/hr → ft/sec → m/s
Master Unit Cancellation to Excel in Problem Solving!

Questions & Answers

Question: How do I know which conversion factor to put in the numerator versus the denominator?

Answer: The key is to place the conversion factor so that the unit you want to eliminate appears in both the numerator and denominator, allowing it to cancel out.

Example: Converting 5 miles to feet (1 mile = 5280 feet)

  • Start with: 5 miles
  • Want to eliminate miles, so place miles in the denominator: \(\frac{5280 \text{ feet}}{1 \text{ mile}}\)
  • Calculation: \(5 \text{ miles} \times \frac{5280 \text{ feet}}{1 \text{ mile}} = 26400 \text{ feet}\)

The miles cancel out, leaving feet. Always arrange conversions so unwanted units cancel!

Question: What if I need to convert multiple units in one problem?

Answer: Convert one unit at a time sequentially. You can multiply all conversion factors together:

Example: Convert 60 mph to ft/sec

\(60 \frac{\text{miles}}{\text{hour}} \times \frac{5280 \text{ ft}}{1 \text{ mile}} \times \frac{1 \text{ hour}}{3600 \text{ sec}}\)

This equals: \(\frac{60 \times 5280}{3600} = 88 \frac{\text{ft}}{\text{sec}}\)

The miles cancel with miles, hours cancel with hours, leaving feet per second.

Work systematically through each unit that needs conversion!

Question: How can I verify that my unit conversion is correct?

Answer: There are several verification methods:

  1. Unit check: Verify that the final units match what you wanted
  2. Magnitude check: See if the numerical value makes sense (e.g., feet should be a larger number than miles for the same distance)
  3. Reverse conversion: Convert back to the original units to see if you get the original value
  4. Estimation: Use approximate values to see if your answer is reasonable

Example: 1 mile = 5280 feet

If you converted 2 miles to feet and got 10560 feet, verify: 10560 ÷ 5280 = 2 miles ✓

The reverse conversion confirms your answer is correct!