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Chapter 1: Problem 3

(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 \(\mathrm{mL}\) ). Show that a milliliter is the same as a cubic centimeter. (c) How many cubic centimeters of water are there in a \(1.00 \mathrm{~L}\) bottle of drinking water?

Short Answer

Expert verified
(a) 1 mile 鈮� 1.609 km. (b) 1 mL = 1 cm鲁. (c) 1 L = 1000 cm鲁.

Step by step solution

01

Understanding the Conversion for Miles to Kilometers

We know from the problem statement and general knowledge that 1 inch is equal to 2.54 centimeters. We need to convert 1 mile to kilometers. The relationships we need are: - 1 mile = 5280 feet, - 1 foot = 12 inches, - 1 inch = 2.54 cm, - 100,000 cm = 1 km. We'll use these sequentially to convert units from miles to kilometers step by step.
02

Converting Miles to Inches

First, convert 1 mile into inches using the intermediate step of converting to feet: 1 mile = 5280 feet 1 foot = 12 inches So, 1 mile = 5280 feet 脳 12 inches/foot = 63360 inches.
03

Converting Inches to Centimeters

Next, convert inches to centimeters. Since 1 inch equals 2.54 centimeters, we apply this conversion: 1 mile = 63360 inches 脳 2.54 cm/inch = 160934.4 cm.
04

Converting Centimeters to Kilometers

Finally, convert centimeters to kilometers knowing that 100,000 centimeters are equivalent to 1 kilometer: 160934.4 cm 梅 100,000 cm/km 鈮� 1.609344 km. Thus, 1 mile 鈮� 1.609 kilometers.
05

Understanding 1 mL Equals 1 Cubic Centimeter

A milliliter (mL) is a unit of volume equivalent to 1 cubic centimeter. This is derived from the definition of 1 liter being equal to 1000 milliliters and simultaneously equal to 1000 cubic centimeters. Hence, 1 mL = 1 cm鲁.
06

Converting Liters to Cubic Centimeters

1 liter is defined as the volume of a cube that is 10 cm on each side (10 cm 脳 10 cm 脳 10 cm = 1000 cm鲁). Thus, a 1-liter bottle of water contains 1000 cubic centimeters. Therefore, 1 L = 1000 cm鲁.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Miles to Kilometers
Converting miles to kilometers is a common task in understanding different measurement systems. In the original solution, we see how 1 mile is first converted to inches, then to centimeters, and finally to kilometers. Here's a simplified breakdown:
  • 1 mile is equivalent to 5280 feet.
  • Each foot contains 12 inches, so a mile contains 63360 inches (5280 feet 脳 12 inches/foot).
  • An inch measures 2.54 centimeters.
  • Thus, converting 63360 inches results in 160934.4 centimeters (63360 inches 脳 2.54 cm/inch).
  • There are 100,000 centimeters in a kilometer, so when converted to kilometers, it becomes approximately 1.609 kilometers.
All these steps demonstrate how to faithfully translate a mile into kilometers using basic unit conversions. This approach is crucial when you need to switch between the customary units used in different countries.
Cubic Centimeters
Cubic centimeters (cm鲁) are often used to measure volume, particularly in science and medicine. A cubic centimeter is the volume of a cube with each side measuring 1 centimeter. Its use is prevalent because it provides an efficient measure for small volumes. In simpler terms:
  • 1 cubic centimeter is equivalent to 1 mL (milliliter).
  • This means a cube that is 1 cm on each side has the same volume as 1 mL of liquid.
This equivalence makes cubic centimeters a handy unit for diverse applications, like packing medicine doses or calculating the volume of small objects.
Volume Conversion
Volume conversion involves changing the unit of measurement from one type to another, for instance, from mL to cm鲁. Given the relationship that 1 mL is equal to 1 cm鲁, the conversion between these two units is quite straightforward and requires no calculation. For other measurements, you might use ratios or multiplication factors to achieve the conversion. Let's enhance understanding by illustrating:
  • Converting gallons to liters involves knowing that 1 gallon is approximately 3.785 liters.
  • To adapt volume measures for cooking from cups to milliliters, remember that 1 cup equals about 237 mL.
Understanding these conversions ensures accuracy, especially in contexts like cooking, scientific experiments, and industry processes.
Liter to Cubic Centimeter
Converting liters to cubic centimeters is a straightforward process due to their defined mathematical relationship:
  • 1 liter is equivalent to 1000 cubic centimeters.
  • This is because a liter represents the volume of a cube that is 10 cm on each side (10 cm 脳 10 cm 脳 10 cm = 1000 cm鲁).
Thus, when you come across a bottle of water labeled as 1 liter, you can be confident it contains 1000 cubic centimeters of water. This relationship not only simplifies conversions but also helps in easily grasping volume comparison in everyday measurements.
Measurement System Conversion
Measurement system conversion is an essential skill for interpreting data across different unit systems. It involves translating one set of units into another, enabling uniformity and comprehension across global metrics. Key considerations include:
  • Familiarize yourself with common conversion factors, like 1 inch = 2.54 cm or 1 pound = 0.453592 kilograms.
  • Understand how to apply these conversions systematically in calculations, ensuring each step maintains accuracy.
  • Practice using conversion tools or calculators for complex conversions, especially when dealing with compounded units like speed (miles per hour to kilometers per hour).
By mastering these concepts, you become adept at moving fluidly between systems like the Imperial and Metric, breaking down potential communication barriers.

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Most popular questions from this chapter

A plane leaves Seattle, flies \(85 \mathrm{mi}\) at \(22^{\circ}\) north of east, and then changes direction to \(48^{\circ}\) south of east. After flying \(115 \mathrm{mi}\) in this new direction, the pilot must make an emergency landing on a field. The Seattle airport facility dispatches a rescue crew. (a) In what direction and how far should the crew fly to go directly to the field? Use components to solve this problem. (b) Check the reasonableness of your answer with a careful graphical sum.

Some commonly occurring quantities. All of the quantities that follow will occur frequently in your study of physics. (a) Express the speed of light \(\left(3.00 \times 10^{8} \mathrm{~m} / \mathrm{s}\right)\) in \(\mathrm{mi} / \mathrm{s}\) and \(\mathrm{mi} / \mathrm{h} .\) (b) Find the speed of sound in air at \(0^{\circ} \mathrm{C}(1100 \mathrm{ft} / \mathrm{s})\) in \(\mathrm{m} / \mathrm{s}\) and \(\mathrm{mi} / \mathrm{h} .\) (c) Show that \(60 \mathrm{mi} / \mathrm{h}\) is the same as \(88 \mathrm{ft} / \mathrm{s}\). (d) Convert the acceleration of a freely falling body \(\left(9.8 \mathrm{~m} / \mathrm{s}^{2}\right)\) to \(\mathrm{ft} / \mathrm{s}^{2}\).

Although these quantities vary from one type of cell to another, a cell can be \(2.0 \mu \mathrm{m}\) in diameter with a cell wall 50.0\(\mathrm{nm}\) thick. If the density (mass divided by volume) of the wall material is the same as that of pure water, what is the mass (in mg) of the cell wall, assuming the cell to be spherical and the wall to be a very thin spherical shell?

While driving in an exotic foreign land, you see a speed-limit sign on a highway that reads 180,000 furlongs per fortnight. How many miles per hour is this? (One furlong is \(\frac{1}{8}\) mile, and a fortnight is 14 days. A furlong originally referred to the length of a plowed furrow.)

I A disoriented physics professor drives \(3.25 \mathrm{~km}\) north, then \(4.75 \mathrm{~km}\) west, and then 1.50 \(\mathrm{km}\) south. (a) Use components to find the magnitude and direction of the resultant displacement of this professor. (b) Check the reasonableness of your answer with a graphical sum.
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