Problem 56 - Chemistry: The Central Science in SI Units, Global Edition

Question

(a) The speed of light in a vacuum is \(2.998 \times 10^{8} \mathrm{~m} / \mathrm{s}\). Calculate its speed in miles per hour. (b) The Sears Tower in Chicago is \(1454 \mathrm{ft}\) tall. Calculate its height in meters. \((\mathbf{c})\) 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 \(242 \mathrm{mg}\) 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?

Step-by-Step Solution

Answer

The short answers for each part are: (a) The speed of light in a vacuum in miles per hour is approximately \(670,616,629\) mph. (b) The height of the Sears Tower in Chicago in meters is approximately \(443\) meters. (c) The volume of the Vehicle Assembly Building in standard exponential notation is \(3.6665 \times 10^9\) liters. (d) The individual's total blood cholesterol is approximately \(125.44\) grams.

Step 1: Given speed

The speed of light in a vacuum is given as \(2.998 \times 10^{8} \mathrm{~m} / \mathrm{s}\).

Step 2: Conversion factors

We need to convert meters to miles and seconds to hours. The conversion factors are: 1 mile = 1609.34 meters, and 1 hour = 3600 seconds.

Step 3: Apply conversion factors

To find the speed of light in miles per hour, we can multiply the given speed by the conversion factors: \[\frac{2.998 \times 10^8 \, \text{m}}{\text{s}} \times \frac{1 \, \text{mile}}{1609.34 \, \text{m}} \times \frac{3600 \, \text{s}}{1 \, \text{h}}\]

Step 4: Calculate the speed in miles per hour

Now, we can perform the calculations and cancel out the units: \[\frac{2.998 \times 10^8}{1609.34} \times 3600 \, \text{mph} \approx 670,616,629 \, \text{mph}\] The speed of light in a vacuum in miles per hour is approximately 670,616,629 mph. #Exercise (b)#: Convert the height of the Sears Tower in Chicago from feet to meters.

Step 1: Given height

The height of the Sears Tower in Chicago is given as 1454 ft.

Step 2: Conversion factor

We need to convert feet to meters. The conversion factor is: 1 meter = 3.281 feet.

Step 3: Apply conversion factor

To find the height in meters, we can multiply the given height by the conversion factor: \[\frac{1454 \, \text{ft}}{1} \times \frac{1 \, \text{m}}{3.281 \, \text{ft}}\]

Step 4: Calculate the height in meters

Perform the calculation and cancel out the units: \[\frac{1454}{3.281} \, \text{m} \approx 443 \, \text{m}\] The height of the Sears Tower in Chicago in meters is approximately 443 meters. #Exercise (c)#: Convert the volume of the Vehicle Assembly Building at the Kennedy Space Center in Florida from cubic meters to liters in standard exponential notation.

Step 1: Given volume

The volume of the Vehicle Assembly Building is given as \(3,666,500 \mathrm{~m}^{3}\).

Step 2: Conversion factor

We need to convert cubic meters to liters. The conversion factor is: 1 cubic meter = 1000 liters.

Step 3: Apply conversion factor

To find the volume in liters, we can multiply the given volume by the conversion factor: \[3,666,500 \, \text{m}^3 \times \frac{1000 \, \text{L}}{1 \, \text{m}^3}\]

Step 4: Calculate the volume in liters

Perform the calculation: \[3,666,500 \times 1000 \, \text{L} = 3.6665 \times 10^9 \, \text{L}\] The volume of the Vehicle Assembly Building in standard exponential notation is \(3.6665 \times 10^9\) liters. #Exercise (d)#: Find the total grams of blood cholesterol in the individual's body.

Step 1: Given cholesterol concentration and blood volume

The cholesterol concentration is given as 242 mg per 100 mL of blood. The total blood volume of the individual is 5.2 L.

Step 2: Conversion factor

Convert blood volume to milliliters: 1 L = 1000 mL.

Step 3: Apply conversion factor

Convert the blood volume from liters to milliliters: \[5.2 \, \text{L} \times \frac{1000 \, \text{mL}}{1 \, \text{L}} = 5200 \, \text{mL}\] So, the total blood volume is 5200 mL.

Step 4: Calculate total cholesterol in mg

To find the total cholesterol in mg, multiply the concentration by the blood volume: \[\frac{242 \, \text{mg}}{100 \, \text{mL}} \times 5200 \, \text{mL} = 125440 \, \text{mg}\] The total blood cholesterol in the individual's body is 125,440 mg.

Step 5: Convert mg to grams

Convert the total blood cholesterol in mg to grams using the conversion factor: 1 g = 1000 mg. \[\frac{125440 \, \text{mg}}{1} \times \frac{1 \, \text{g}}{1000 \, \text{mg}} = 125.44 \, \text{g}\] The individual's total blood cholesterol is approximately 125.44 grams.

Key Concepts

Speed of Light

Understanding how fast light travels can truly blow your mind! Light zips around at an incredible speed, and in its purest form, it moves at approximately \(2.998 \times 10^8\) meters per second in a vacuum. This is often abbreviated as \(c\), the speed of light. However, when we talk about this speed in the context of everyday measurements, like miles per hour, we need to make some conversions.
To convert the speed of light into miles per hour, we have to do two things:

• Change meters into miles using the fact that 1 mile is approximately 1609.34 meters.
• Convert seconds into hours since 1 hour is 3600 seconds.

When we plug those conversion factors into our equation, we get that the speed of light in miles per hour is roughly 670,616,629 mph. This gargantuan number highlights just how extraordinarily fast light travels—it could circle the earth's equator over seven times in one second! This conversion from metric to imperial units is a handy trick whenever you need to understand measurements in different contexts.

Volume Conversion

Measuring volume, especially in large structures, often requires conversions to make sense of the numbers. A fascinating example is the Vehicle Assembly Building at the Kennedy Space Center. Its volume is given as 3,666,500 cubic meters. But what if we wanted to express this enormous volume in liters instead?
Conversion from cubic meters to liters is quite straightforward since 1 cubic meter is equal to 1000 liters. By multiplying the given volume by 1000, you switch from cubic meters to liters.
Following this, the volume of this massive building is calculated to be \(3.6665 \times 10^9\) liters when expressed in exponential notation. Exponential notation is a tidy way to express large numbers like this one, emphasizing how vast the space is inside that building. Converting such measurements can provide a clearer picture, especially if you are more familiar with one unit than another.

Blood Cholesterol Measurement

Blood cholesterol is a common and essential health parameter to monitor, presented as milligrams of cholesterol per 100 milliliters of blood. In this example, someone has a cholesterol level of 242 mg/100 mL, and they have a total blood volume of 5.2 liters.
To find out the total cholesterol in their body, we first convert the blood volume from liters to milliliters. Since there are 1000 milliliters in a liter, multiplying 5.2 liters by 1000 gives us 5200 milliliters.
Once we have the total volume in a universal metric, we can calculate the total cholesterol by applying the known concentration: multiply 242 mg/100 mL by the total 5200 mL to find the cholesterol content. This comes out to 125,440 mg.
Finally, converting this figure to grams by dividing by 1000 (since 1 gram equals 1000 milligrams), results in a total cholesterol amount of approximately 125.44 grams. Understanding these conversion steps can be crucial for interpreting medical results in a more comprehensible way, reinforcing the importance of accurate health measurements.