Question

Which of the following statements is (are) true? a. A 1.00 -liter bottle contains more soda than a 1.00 -quart bottle. b. A man who is \(1.52 \mathrm{~m}\) tall is taller than a woman who is \(5 \mathrm{ft} 3 \mathrm{in}\). tall. c. A 335 -g container of peanut butter is heavier than a container holding \(\frac{1}{2}\) pound. d. A bus moving at a speed of \(45 \mathrm{mi} / \mathrm{hr}\) is traveling faster than a car moving at \(65 \mathrm{~km} / \mathrm{hr}\)

Short answer

Statements A, C, and D are true. Statement B is false.

Step-by-step solution

Convert Between Units - Statement A

First, let's convert liters to quarts using the conversion factor 1 liter ≈ 1.057 quarts. The 1.00-liter bottle can be converted to \(1.00 * 1.057 = 1.057\) quarts. Now, let's compare the two amounts: 1.00 liter = 1.057 quarts 1.00 quart = 1.00 quart Since 1.057 quarts > 1.00 quart, statement A is true.

Convert Between Units - Statement B

To compare the heights of the man and woman, we'll need to convert both heights to the same unit. For the woman's height, convert 5 feet and 3 inches to meters. 5 feet = 5 * 0.3048 = 1.524 meters 3 inches = 3 * 0.0254 = 0.0762 meters The woman's height = 1.524 meters + 0.0762 meters = 1.6002 meters Now compare the heights: Man's height = 1.52 meters Woman's height = 1.6002 meters Since 1.52 meters < 1.6002 meters, statement B is false.

Convert Between Units - Statement C

Convert the weight of the peanut butter containers to the same units. The 335-g container can be converted to pounds using the conversion factor 1 gram ≈ 0.00220462 pounds. 335 grams = 335 * 0.00220462 ≈ 0.7385 pounds Now compare the weights: 335-g container = 0.7385 pounds ½-pound container = 0.5 pounds Since 0.7385 pounds > 0.5 pounds, statement C is true.

Convert Between Units - Statement D

To compare the speeds of the bus and car, we'll need to convert one speed to the same units as the other. We can convert miles per hour to kilometers per hour using the conversion factor: 1 mile ≈ 1.60934 kilometers. The bus's speed can be converted to kilometers per hour: 45 miles/hr ≈ 45 * 1.60934 ≈ 72.4203 km/hr Now compare the speeds: Bus's speed ≈ 72.4203 km/hr Car's speed = 65 km/hr Since 72.4203 km/hr > 65 km/hr, statement D is true. In conclusion, statements A, C, and D are true. Statement B is false.

Key concepts

Measurement Conversion

Measurement conversion is essential when comparing quantities from different systems of measurement. It allows you to express a quantity in a different unit without changing its value. For instance, converting liters to quarts involves using a conversion factor that links the two units.

Conversion factors are ratios derived from the equality between two different units. In the exercise, you see conversions like:

• 1 liter to quarts using 1 liter ≈ 1.057 quarts.
• Feet and inches to meters using 1 foot ≈ 0.3048 meters and 1 inch ≈ 0.0254 meters.
• Grams to pounds using 1 gram ≈ 0.00220462 pounds.
• Miles to kilometers using 1 mile ≈ 1.60934 kilometers.

These conversions allow for meaningful comparisons whether you're discussing volume, height, weight, or speed.

Metric System

The metric system is a decimal-based system of measurement used worldwide. It simplifies calculations and conversions because it's based on powers of ten. Units in the metric system include meters for length, liters for volume, and grams for weight. In the conversion examples above, you saw the use of meters, liters, and grams.

One of the advantages of the metric system is its coherence and ease of use across different scientific and everyday contexts. This system often requires fewer conversion steps, and its units are interrelated (e.g., 1 liter is equivalent to 1,000 milliliters).

• The man's height of 1.52 meters is converted directly or easily compared to the woman’s height in meters.
• Volume conversions from liters are straightforward due to direct relationships.

This consistency makes the metric system particularly advantageous in science, engineering, and everyday use.

Imperial System

The imperial system, also known as the British Imperial System, is less straightforward than the metric system as it isn't based on multiples of ten. This can make conversions more complex. Common imperial units include feet and inches for length, pounds for weight, and miles for distance.

In the exercise, you find conversions like:

• The woman's height in feet and inches converted into meters to provide a comparison with the man's height.
• The half-pound container compared against a gram measurement.

The imperial system is prevalent primarily in the United States for everyday measurements. Its historical roots make it common, though it presents challenges in scientific contexts due to its less linear conversion ratios.

Comparison of Units

Comparing units from different measuring systems requires conversion and understanding of relative magnitudes. In the exercise example, understanding which units to convert into can help make accurate comparisons. Each conversion answers whether the statement in question is true or false.

For example:

• The volume comparison shows that 1 liter is larger than 1 quart when converted properly.
• By converting feet and inches to meters, the precise comparison shows the woman's height exceeds the man's.
• Weight comparison in pounds versus grams determines that 335 grams is more than half a pound when using correct conversion factors.
• The bus's speed, when converted from miles per hour to kilometers per hour, exceeds the car's speed, solving the comparison.

These examples demonstrate how conversion is pivotal, ensuring different systems remain useful and comparable.