Question

(a) The speed of light in a vacuum is \(2.998 \times 10^{8} \mathrm{~m} / \mathrm{s}\). Calculate its speed in \(\mathrm{km} / \mathrm{hr} .\) (b) The Sears Tower in Chicago is \(1454 \mathrm{ft}\) tall. Calculate its height in meters. (c) The Vehicle Assembly Building at the Kennedy Space Center in Florida has a volume of \(3,666,500 \mathrm{~m}^{3}\). Convert this volume to liters, and express the result in standard exponential notation. (d) An individual suffering from a high cholesterol level in her blood has \(232 \mathrm{mg}\) of cholesterol per \(100 \mathrm{~mL}\) of blood. If the total blood volume of the individual is \(5.2 \mathrm{~L}\), how many grams of total blood cholesterol does the individual's body contain?

Short answer

(a) \(1.079 \times 10^{9}\ \mathrm{km/hr}\) (b) \(443.1864\ \mathrm{m}\) (c) \(3.6665 \times 10^9\ \mathrm{L}\) (d) \(12.064\ \mathrm{g}\)

Step-by-step solution

(a) Convert speed of light from meters/second to kilometers/hour

First, note that there are 3.6 kilometers in an hour for every meter per second. To find the speed of light in kilometers/hour, multiply the given speed by the conversion factor: \(speed_{km/hr} = speed_{m/s} \times \frac{3.6\ \mathrm{km/hr}}{1\ \mathrm{m/s}}\)

(a) Calculating speed of light in kilometers/hour

Using the conversion factor, we can now calculate the speed of light in kilometers/hour: \(speed_{km/hr} = (2.998 \times 10^{8}\ \mathrm{m/s}) \times \frac{3.6\ \mathrm{km/hr}}{1\ \mathrm{m/s}} = 1.079 \times 10^{9}\ \mathrm{km/hr}\)

(b) Convert Sears Tower height from feet to meters

To convert the height of the Sears Tower from feet to meters, we need the conversion factor, which is 1 foot equals 0.3048 meters: \(height_{m} = height_{ft} \times \frac{0.3048\ \mathrm{m}}{1\ \mathrm{ft}}\)

(b) Calculating Sears Tower height in meters

Using the conversion factor, we can now calculate the height of the Sears Tower in meters: \(height_{m} = 1454\ \mathrm{ft} \times \frac{0.3048\ \mathrm{m}}{1\ \mathrm{ft}} = 443.1864\ \mathrm{m}\)

(c) Convert Vehicle Assembly Building volume to liters and express in exponential notation

To convert the volume of the Vehicle Assembly Building from cubic meters to liters, we use the conversion factor that there are 1000 liters in 1 cubic meter: \(volume_{L} = volume_{m^3} \times \frac{1000\ \mathrm{L}}{1\ \mathrm{m^3}}\)

(c) Calculating Vehicle Assembly Building volume in liters and exponential notation

Using the conversion factor, we can now calculate the volume of the Vehicle Assembly Building in liters and convert to exponential notation: \(volume_{L} = 3666500\ \mathrm{m^3} \times \frac{1000\ \mathrm{L}}{1\ \mathrm{m^3}} = 3.6665 \times 10^9\ \mathrm{L}\)

(d) Calculate cholesterol concentration in grams per mL

In order to find the total grams of cholesterol in the individual's blood, we need to convert the given concentration to grams/mL: \(concentration_{g/mL} = concentration_{mg/mL} \times \frac{1\ \mathrm{g}}{1000\ \mathrm{mg}}\)

(d) Calculating cholesterol concentration in grams per mL

Using the conversion factor, we can now calculate the cholesterol concentration in grams/mL: \(concentration_{g/mL} = \frac{232\ \mathrm{mg}}{100\ \mathrm{mL}} \times \frac{1\ \mathrm{g}}{1000\ \mathrm{mg}} = 0.00232\ \mathrm{g/mL}\)

(d) Calculate total grams of cholesterol in the individual's blood

To find the total grams of cholesterol in the individual's blood, multiply the cholesterol concentration in grams/mL by the total blood volume in liters (1 L = 1000 mL): \(total\ cholesterol_{g} = concentration_{g/mL} \times total\ volume_{mL} = 0.00232\ \mathrm{g /mL} \times (5.2\ \mathrm{L} \times 1000\ \mathrm{mL /L})\)

(d) Calculating total grams of cholesterol in the individual's blood

Now, let's calculate the final result for the total grams of cholesterol in the individual's blood: \(total\ cholesterol_{g} = 0.00232\ \mathrm{g/mL} \times 5200\ \mathrm{mL} = 12.064\ \mathrm{g}\) In summary, the results are as follows: (a) Speed of light: \(1.079 \times 10^{9}\ \mathrm{km/hr}\) (b) Sears Tower height: \(443.1864\ \mathrm{m}\) (c) Vehicle Assembly Building volume: \(3.6665 \times 10^9\ \mathrm{L}\) (d) Total cholesterol in the individual's blood: \(12.064\ \mathrm{g}\)

Key concepts

Speed of Light Conversion

Understanding how to convert the speed of light from one unit to another is crucial in various fields, including physics and astronomy. The speed of light, typically measured in meters per second (m/s), can be expressed in different units such as kilometers per hour (km/hr) to provide a sense of scale that might be more familiar. The conversion process involves multiplying the speed value by a factor that accounts for the differences in units.

In the provided exercise solution, a conversion factor of 3.6 is used since there are 3600 seconds in an hour and 1000 meters in a kilometer, making the factor simply 3600/1000 or 3.6. Therefore, to convert from m/s to km/hr, you multiply by 3.6. This straightforward approach helps to simplify what could otherwise be a complex conversion.

Height Measurement Conversion

Conversions between the imperial system (feet, inches) and the metric system (meters) are common in height measurement conversions. When converting from feet to meters, as with the height of a building like the Sears Tower, it's important to use the exact conversion factor of 1 foot being equivalent to 0.3048 meters.

The solution to the exercise uses this conversion factor to change the measurement from feet to meters by multiplying the number of feet by 0.3048. This type of conversion is not only useful in academic settings but also in practical situations, such as construction or international travel where measurements may need to be understood or communicated in differing units.

Volume Conversion to Liters

When dealing with volume conversions, it's often necessary to convert from one unit of volume to another, such as from cubic meters (m³) to liters (L). This conversion is particularly relevant in chemistry and other scientific fields where the volume of liquids or gases is frequently measured in liters for practical purposes.

The solution to the problem uses the direct relation that 1 cubic meter equals 1000 liters. By multiplying the volume in cubic meters by 1000, we obtain the volume in liters. Expressing the result in standard exponential notation, as done in the problem, is a convenient way of representing large numbers and is a common convention in scientific communication.

Blood Cholesterol Level Calculation

Calculating blood cholesterol levels is an essential task in healthcare, as it provides information about a person's risk for heart disease. Measurement units for cholesterol levels often include milligrams of cholesterol per deciliter of blood (mg/dL). However, the total amount of cholesterol in the body is more commonly expressed in grams.

To find the total cholesterol level, the first step is to convert the concentration from mg/mL to g/mL using the conversion factor of 1 gram being equivalent to 1000 milligrams. After applying this factor, the total blood volume (in liters, where 1 liter = 1000 mL) is multiplied by the converted concentration to get the total cholesterol in grams. This two-step conversion process is critical for arriving at a clear and understandable result that can inform medical decisions.