Problem 4 - College Physics

Question

\(\bullet\) (a) Starting with the definition 1.00 in. \(=2.54 \mathrm{cm},\) find the number of kilometers in 1.00 mile. (b) In medicine, volumes are often expressed in milliliters (ml or mL). Show that a milliliter is the same as a cubic centimeter. (c) How many cubic centimeters of water are there in a 1.00 L bottle of drinking water?

Step-by-Step Solution

Answer

1 mile = 1.609344 km; 1 mL = 1 cm³; 1 L = 1000 cm³.

Step 1: Convert inches to centimeters

We start with the definition 1.00 inch = 2.54 cm. To find the number of kilometers in a mile, we first convert miles to inches using the fact that 1 mile = 5280 feet and 1 foot = 12 inches. Thus, 1 mile = 5280 feet × 12 inches/foot = 63360 inches. Then, we convert inches to centimeters: 63360 inches × 2.54 cm/inch = 160934.4 cm.

Step 2: Convert centimeters to kilometers

Next, we convert centimeters to kilometers. We know that 1 kilometer = 100000 cm. Therefore, we can convert centimeters to kilometers using the conversion: 160934.4 cm ÷ 100000 cm/km = 1.609344 km. So, there are 1.609344 kilometers in a mile.

Step 3: Define milliliter in terms of cubic centimeters

A milliliter (mL) is equivalent to a cubic centimeter (cc). This is because 1 mL is defined as the volume of a cube with 1 cm on each side. Thus, 1 mL = 1 cm³.

Step 4: Convert liters to cubic centimeters

Since 1 liter (L) is defined as 1000 milliliters, and since 1 milliliter is the same as 1 cubic centimeter (as established in Step 3), we can say that 1.00 L = 1000 mL = 1000 cm³.

Key Concepts

Distance Conversion

When converting distances, it’s important to know the relationships between various measurement units. Converting from inches to centimeters, for instance, is straightforward because 1 inch is exactly 2.54 centimeters. This fixed conversion factor can be used universally to switch between the two units. To convert larger distances like miles into kilometers, more steps and conversions are needed.
If you start with 1 mile, remember that one mile is equivalent to 5280 feet, and each foot contains 12 inches. Therefore, 1 mile equals 63360 inches (5280 feet x 12 inches/foot).

• Inches are converted to centimeters by multiplying by the conversion factor of 2.54 cm/inch.
• Then, to convert from centimeters to kilometers, simply divide by 100,000 (because there are 100,000 centimeters in a kilometer).

This guides us in understanding and executing distance conversions efficiently across different units.

Volume Conversion

Volume conversion involves changing measurements from one volume unit to another. It's common in everyday life, especially in fields like cooking or science. For liquids, milliliters (mL) and liters are frequently used metric units.

• 1 milliliter (mL) is equivalent to 1 cubic centimeter (cc), which is important for understanding various scientific contexts.
• To convert liters to cubic centimeters, remember that 1 liter is composed of 1000 milliliters.

This highlights that volume conversion within the metric system is relatively simple because it's based on powers of 10, easing calculations.

Metric System

The metric system, used by most of the world, is a decimal-based system of measurement. It simplifies calculations and conversions due to its uniform structure.

• Commonly used metric units include meters for distance, liters for volume, and grams for weight.
• The system revolves around base units multiplied or divided by powers of ten, e.g., milli (1/1000), centi (1/100), and kilo (1000).

This organization means converting within the metric system is typically a straightforward process, involving simple shifting of the decimal point.

Cubic Centimeters

Cubic centimeters (cm³), often referred to as cc, are a unit of volume used within the metric system. This unit is widely used in fields like medicine and automotive to describe engine displacement.

• A cubic centimeter measures the space occupied by a cube with 1 cm on each side.
• This volume unit is directly comparable to a milliliter, as 1 cubic centimeter equals 1 milliliter.

Understanding cubic centimeters is crucial for accurate volume measurements in scientific and engineering applications, ensuring precision in data collection and application.