Make the conversion indicated in each of the following: (a) the men's world record long jump, 29 ft \(4 \%\) in., to meters (b) the greatest depth of the ocean, about \(6.5 \mathrm{mi}\), to kilometers (c) the area of the state of Oregon, \(96,981 \mathrm{mi}^{2}\), to square kilometers (d) the volume of 1 gill (exactly 4 oz) to milliliters (e) the estimated volume of the oceans, \(330,000,000 \mathrm{mi}^{3},\) to cubic kilometers. (f) the mass of a 3525 -lb car to kilograms (g) the mass of a 2.3 -oz egg to grams

Understanding the conversion of units is critical in fields like chemistry, physics, engineering, and everyday life. It involves changing a quantity expressed in one set of units to another set of units without changing the magnitude of the quantity. For example, converting a distance from feet to meters or a volume from gallons to liters.

When you tackle a conversion problem, like changing the men's world record long jump from feet and inches to meters, you start by expressing the entire measurement in a common unit (in this case, inches). This makes it easier to apply the conversion factor—the specific value you multiply by—to get the desired units (meters in this example). Always keep an eye on your units throughout the problem to ensure accuracy.

Dimensional analysis is the process by which you can convert one set of units to another, using multiplication or division by conversion factors. This technique guarantees that the final units are what you need, because only like units cancel out, leaving the desired units in your result.

In dimensional analysis, you follow a systematic approach: set up the initial quantity with its units, multiply by conversion factors where units cancel out from numerator and denominator until the desired units remain. It's a powerful tool for solving exercises like converting the estimated volume of the oceans from cubic miles to cubic kilometers.

Key Steps in Dimensional Analysis:

• Write down the quantity to convert with its unit.
• Set up conversion factors as fractions, where the unit you want to eliminate is opposite of the unit in your quantity.
• Multiply the fractions, canceling out the units until the desired unit is left.
• Calculate the final value, ensuring the unit consistency and correctness.

The metric system is an international system of units (SI) based on multiples of ten, making it simpler to understand and convert between units. For instance, the basic unit of length is the meter. Other units are derived from it by adding prefixes that represent powers of ten, such as centimeters (100th of a meter) and kilometers (1000 meters).

Chemistry heavily relies on this system for measuring volume, length, mass, and temperature. Understanding the metric system is vital when you're asked to convert the mass of a 2.3-oz egg to grams because you're working within the metric system's framework for mass.

The imperial system, used mainly in the United States, consists of units such as pounds, inches, and gallons. Converting imperial units to metric units, which is used by most of the world, involves using specific conversion factors.

For example, to convert the mass of a car from pounds to kilograms, identify the conversion factor (1 pound equals 0.453592 kilograms) and use it to multiply the value in pounds. Similarly, to convert fluid ounces to milliliters, use the conversion factor that 1 US fluid ounce equals approximately 29.5735 milliliters. Such conversions are crucial in science and international trade, where consistent and precise measurements are necessary.