Question

Carry out these conversions: (a) A 6.0-ft person weighs 168 lb. Express this person's height in meters and weight in kilograms. \((1 \mathrm{lb}=453.6 \mathrm{~g} ; 1 \mathrm{~m}=\) \(3.28 \mathrm{ft} .)\) (b) The current speed limit in some states in the United States is 55 miles per hour. What is the speed limit in kilometers per hour? (c) The speed of light is \(3.0 \times 10^{10} \mathrm{~cm} / \mathrm{s}\). How many miles does light travel in 1 hour? (d) Lead is a toxic substance. The "normal" lead content in human blood is about 0.40 part per million (that is, \(0.40 \mathrm{~g}\) of lead per million grams of blood). A value of 0.80 part per million (ppm) is considered to be dangerous. How many grams of lead are contained in \(6.0 \times 10^{3} \mathrm{~g}\) of blood (the amount in an average adult) if the lead content is \(0.62 \mathrm{ppm} ?\)

Short answer

\n (a) The height of the person is approximately 1.83 meters and weight is around 76.20 kilograms.\n(b) The speed limit in kilometers per hour is approximately 88.51 km/h.\n(c) In one hour, light travels about \(2.99792458 * 10^9\) miles.\n(d) In \(6.0 * 10^3\) grams of blood, there are approximately \(0.00372\) grams of lead.

Step-by-step solution

- Conversion from feet to meter and pound to kilograms

Calculate the person's height in meters by multiplying the height in feet (6.0 ft) by the conversion factor \(1 m/3.28 ft\). In a similar manner, calculate the weight of the person in kilograms by multiplying weight in pounds by the conversion factor \(0.4536 kg/lb.\)\nHeight in meters = 6.0 ft * \(1 m/3.28 ft\).\nWeight in kg = 168lb * \(0.4536 kg/lb\) .

- Conversion from miles per hour to kilometers per hour

Convert the speed limit from miles per hour to kilometers per hour by multiplying the speed limit in miles per hour by \(1.609 km/mile\).\n Speed limit in km/h = 55 miles/hour * \(1.609 km/mile\).

- Conversion of speed of light from cm/sec to miles/hour

Calculate how many miles light travels in one hour by converting the speed of light from \(3*10^10\) cm/s to miles/hour. Use conversion factors \(1 m/100 cm\), \(1 km/1000 m\), \(1 mile/1.609 km\), and \(3600 s/hour\).\nMiles light travels in 1 hour = \(3*10^{10} cm/s * (1 m/100 cm) * (1 km/1000 m) * (1 mile/1.609 km) * (3600 s/hour)\).

- Conversion from ppm to g

Calculate how many grams of lead are contained in \(6.0 * 10^3 g\) of blood using the provided concentration of \(0.62 ppm\).\nGrams of lead in blood = \(6.0*10^3 g\) of blood * \(0.62 g lead)/10^6 g blood\).

Key concepts

Metric Conversion

Understanding metric conversion is essential in various scientific and daily life scenarios. It involves changing one unit of measure to another without changing the quantity. For instance, converting a person's height and weight from the US customary units (feet and pounds) to the metric units (meters and kilograms) requires using specific conversion factors. For height, we multiply the number of feet by the factor of 1 meter per 3.28 feet. Similarly, for weight, we use the factor of 0.4536 kilograms per pound. This straightforward linear transformation allows us to seamlessly switch between different measurement systems.

In our example, a person weighing 168 pounds and standing 6 feet tall can be expressed as 1.83 meters in height and 76.20 kilograms in weight after applying these conversion factors. It's essential to use these factors correctly to obtain accurate results, and remembering these key conversion factors can simplify many such problems.

Scientific Notation

Scientific notation is a convenient way of expressing very large or very small numbers. It simplifies calculations and provides a form that is easier to read and manage. In scientific notation, numbers are written as the product of a number between 1 and 10 and a power of 10. For example, the speed of light can be written as \(3.0 \times 10^{10}\) cm/s, which highlights the magnitude of this incredibly fast speed.

When converting such numbers into different units, we use the same multiplication and division principles that apply to numbers not in scientific notation. It is crucial to keep track of the exponents when converting units and to adjust the power of 10 accordingly. For example, if you were to convert \(3.0 \times 10^{10}\) cm/s to meters per second, you would end up with \(3.0 \times 10^{8}\) m/s, since 100 centimeters make up one meter. Being confident in using scientific notation is invaluable in physics, chemistry, and many technical fields.

Dimensional Analysis

Dimensional analysis, also known as the factor-label method or unit analysis, is a critical tool used to convert one set of units to another. This method revolves around the use of conversion factors, which are ratios that bridge two compatible units. During this process, we systematically cancel out the units we want to convert from and introduce the units we want to convert to.

For example, to convert a speed limit from 55 miles per hour to kilometers per hour, we multiply 55 by the conversion factor of 1.609 kilometers per mile. The miles unit cancels out, and we're left with the speed in kilometers per hour. Likewise, for finding how many grams of lead are in a certain mass of blood with a known concentration in parts per million, we form a ratio that cancels out the unwanted units, leading to the desired result in grams. Mastering dimensional analysis ensures that students can tackle unit conversions with confidence and precision.