Unit Conversion Problems (Challenging)

These problems will challenge your understanding of unit conversions, including those involving areas, volumes, densities, and other derived units. You'll often need to apply conversion factors multiple times.

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Example Problem: Square Inches to Square Meters

Problem: A small solar panel has an area of 150 square inches (in$^2$). What is its area in square meters (m$^2$)?

Step 1: What you have, what you want.

You have 150 in$^2$. You want square meters (m$^2$).

Step 2: Find your helper numbers (conversion factors).

You'll need a way to go from inches to meters. We know:

• 1 inch (in) = 2.54 centimeters (cm)
• 1 meter (m) = 100 centimeters (cm)

Since you're converting *square* inches to *square* meters, you'll need to apply these linear conversion factors twice (or square them).

Step 3: Set it up.

Start with your given value. Then, add your helper numbers, making sure units cancel out. Since it's area (units squared), you need to square the entire conversion factor for length.

$$ 150 \text{ in}^2 \times \left( \frac{2.54 \text{ cm}}{1 \text{ in}} \right)^2 \times \left( \frac{1 \text{ m}}{100 \text{ cm}} \right)^2 $$

This is the same as:

$$ 150 \text{ in}^2 \times \frac{(2.54)^2 \text{ cm}^2}{1^2 \text{ in}^2} \times \frac{1^2 \text{ m}^2}{(100)^2 \text{ cm}^2} $$

Notice how 'in$^2$' and 'cm$^2$' will cancel out, leaving 'm$^2$'.

Step 4: Do the math.

Multiply all the numbers on the top. Then, divide by all the numbers on the bottom. Your answer should have the new units (m$^2$).

$$ 150 \times \frac{(2.54)^2}{1} \times \frac{1}{(100)^2} \text{ m}^2 $$ $$ = 150 \times \frac{6.4516}{1} \times \frac{1}{10000} \text{ m}^2 $$ $$ = 150 \times 0.00064516 \text{ m}^2 \approx 0.096774 \text{ m}^2 $$

Answer: Approximately 0.0968 m$^2$ (rounded to 4 significant figures)

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Common Conversion Factors

Here are some common conversion factors you might need for these problems:

• 1 inch (in) = 2.54 centimeters (cm)
• 1 foot (ft) = 0.3048 meters (m)
• 1 meter (m) = 100 centimeters (cm)
• 1 kilometer (km) = 1000 meters (m)
• 1 mile = 1.609 kilometers (km)
• 1 pound (lb) = 4.448 Newtons (N)
• 1 kilogram (kg) = 1000 grams (g)
• 1 kilogram (kg) = 2.205 pounds (lbs)
• 1 gallon (gal) = 3.785 liters (L)
• 1 second (s) = 1000 milliseconds (ms)
• 1 minute (min) = 60 seconds (s)
• 1 hour (hr) = 60 minutes (min)

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Practice Problems

Solve these problems. Check your answers using the Answer Key at the bottom of the page.

Problem 1: Cubic Feet to Cubic Meters

A large storage container has a volume of 250 cubic feet (ft$^3$). What is its volume in cubic meters (m$^3$)?

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Problem 2: Density Conversion (g/cm³ to kg/m³)

The density of a liquid is 0.85 grams per cubic centimeter (g/cm$^3$). Convert this density to kilograms per cubic meter (kg/m$^3$).

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Problem 3: Pressure Conversion (Pounds per Square Inch to Pascals)

A tire pressure gauge reads 32 pounds per square inch (psi). Convert this pressure to Pascals (Pa), where 1 Pa = 1 Newton per square meter (N/m$^2$).

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Problem 4: Volume Flow Rate (Cubic Feet per Minute to Cubic Meters per Second)

A large industrial fan moves air at a rate of 1500 cubic feet per minute (ft$^3$/min). Convert this flow rate to cubic meters per second (m$^3$/s).

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Show Answer Key

Problem 1: Approximately 7.08 m$^3$

Problem 2: 850 kg/m$^3$

Problem 3: Approximately 220,630 Pa

Problem 4: Approximately 0.708 m$^3$/s

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Related Concepts & Skills

• Unit Conversions (Skill)
• Dimensional Analysis (Concept)