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 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 no- tation. (d) An individual suffering from a high cholesterol level in her blood has 242 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) The speed of light in mph is approximately $6.711\times {10}^8\mathrm{mph}$. (b) The height of the Sears Tower in meters is approximately 443.2 m. (c) The volume of the Vehicle Assembly Building in liters is $3.6665\times {10}^9\mathrm{L}$. (d) The individual's body contains approximately 1258.4 grams of total blood cholesterol.

Step-by-step solution

(a) Convert speed of light to miles per hour

To convert the speed of light from meters per second (m/s) to miles per hour (mph), we will use the following conversion factors: 1 mile = 1609.34 meters 1 hour = 3600 seconds The given speed of light is $2.998\times {10}^8\mathrm{m}/\mathrm{s}$. $Speed(mph)=(2.998\times {10}^8\mathrm{m}/\mathrm{s})\times (\frac{1\mathrm{mile}}{1609.34\mathrm{m}})\times (\frac{3600\mathrm{s}}{1\mathrm{hour}})$ Now, perform the calculations: $Speed(mph)\approx 6.711\times {10}^8\mathrm{mph}$ So, the speed of light in mph is approximately $6.711\times {10}^8\mathrm{mph}$.

(b) Convert Sears Tower height to meters

To convert the height of the Sears Tower from feet to meters, we'll use the following conversion factor: 1 meter = 3.28084 feet The given height of Sears Tower is 1454 ft. $Height(m)=1454ft\times (\frac{1\mathrm{m}}{3.28084\mathrm{ft}})$ Now, perform the calculations: $Height(m)\approx 443.2\mathrm{m}$ So, the height of the Sears Tower in meters is approximately 443.2 m.

(c) Convert Vehicle Assembly Building volume to liters

To convert the volume of the Vehicle Assembly Building from cubic meters (m³) to liters (L), we'll use the following conversion factor: 1 m³ = 1000 L The given volume of the Vehicle Assembly Building is $3,666,500{\mathrm{m}}^3$. $Volume(L)=3,666,500{\mathrm{m}}^3\times (\frac{1,000\mathrm{L}}{1{\mathrm{m}}^3})$ Now, perform the calculations: $Volume(L)=3.6665\times {10}^9\mathrm{L}$ So, the volume of the Vehicle Assembly Building in liters is $3.6665\times {10}^9\mathrm{L}$.

(d) Calculate grams of total blood cholesterol

Given the cholesterol concentration of 242 mg/100 ml and a total blood volume of 5.2 L, we can calculate the total cholesterol in grams by first converting the blood volume to milliliters and then finding the total cholesterol: 1 L = 1000 mL Total blood volume (in mL) = $5.2L\times (\frac{1000\mathrm{mL}}{1\mathrm{L}})=5200\mathrm{mL}$ Now, let's find the total cholesterol in milligrams: Total cholesterol (mg) = $242\frac{\mathrm{mg}}{100\mathrm{mL}}\times 5200\mathrm{mL}$ Now, perform the calculations: Total cholesterol (mg) ≈ 1,258,400 mg Finally, we'll convert this amount to grams: 1 g = 1000 mg Total cholesterol (g) = $1,258,400\mathrm{mg}\times (\frac{1\mathrm{g}}{1000\mathrm{mg}})\approx 1258.4\mathrm{g}$ So, the individual's body contains approximately 1258.4 grams of total blood cholesterol.

Key concepts

Understanding Speed of Light Conversion

Converting the speed of light into different units is essential in physics and engineering. The speed of light is a fundamental constant of the universe at approximately $2.998\times {10}^8\mathrm{m}/\mathrm{s}$. To convert this to miles per hour, which is a common unit of speed in the United States, we need to use two conversion factors relating meters to miles and seconds to hours.

Using $1\mathrm{mile}=1609.34\mathrm{meters}$ and $1\mathrm{hour}=3600\mathrm{seconds}$, we multiply the speed of light by these factors to get the speed in miles per hour. This is a useful conversion when discussing the scale of astronomical distances or the transfer of information in space exploration.

It's important to remember that the speed of light does not change, just the way we measure it does, depending on the units we use.

Height Conversion to Meters

Height conversion is crucial when we need to communicate measurements in a universal unit accepted worldwide, which is the meter. In science and engineering, meters are the standard unit of measurement for length.

For instance, the Sears Tower's height is first given in feet, but converting it to meters provides us with a form that can be easily understood across different geographies. The conversion factor $1\mathrm{meter}=3.28084\mathrm{feet}$ is used to perform this calculation. Knowing how to make such conversions ensures that you can interpret and share measurements confidently in a globally understood language.

Volume Conversion to Liters

Volume conversion to liters is a common task in chemistry, cooking, and industry, as liters are a standard unit for measuring volume. Especially in chemistry, accuracy in volume measurement is critical to ensure the correct proportions in a solution.

To convert from cubic meters, which abbreviate as $m^3$, to liters (L), you use the conversion factor $1{\mathrm{m}}^3=1,000\mathrm{L}$. This simple ratio helps to visualize the size of large structures, like the Volume Assembly Building, in more commonplace units. Knowing this conversion is particularly useful when dealing with measurements related to fluid capacities or storage space.

Cholesterol Level Calculation

Cholesterol level calculation is significant in health and medicine, where understanding and managing these levels can be vital. Cholesterol measurements are usually given in milligrams per deciliter (mg/dL) in medical diagnostics, but they can also be stated per 100 milliliters (mL) of blood. To determine the total amount of cholesterol in an individual's body, you must be able to convert volumes and concentrations into a consistent set of units.

Converting the total blood volume from liters to milliliters and using the given cholesterol concentration allows healthcare professionals to estimate the total cholesterol load. This step is necessary for providing an appropriate medical intervention and recommendations for lifestyle changes.