Problem 55 - Chemistry The Central Science In Si Units

Question

(a) A bumblebee flies with a ground speed of \(15.2 \mathrm{~m} / \mathrm{s}\). Calculate its speed in \(\mathrm{km} / \mathrm{hr}\). (b) The lung capacity of the blue whale is \(5.0 \times 10^{3} \mathrm{~L}\). Convert this volume into gallons. (c) The Statue of Liberty is 151 ft tall. Calculate its height in meters. (d) Bamboo can grow up to \(60.0 \mathrm{~cm} /\) day. Convert this growth rate into inches per hour.

Step-by-Step Solution

Answer

(a) 54.72 km/hr; (b) 1320.86 gallons; (c) 46.02 meters; (d) 0.98425 inches/hour.

Step 1: Convert Speed from m/s to km/hr

To convert speed from meters per second (m/s) to kilometers per hour (km/hr), we use the conversion factors: there are 1000 meters in a kilometer and 3600 seconds in an hour. Therefore, we multiply by 1000 and divide by 3600. So, the speed in km/hr is calculated as:\[15.2 \frac{\mathrm{m}}{\mathrm{s}} \times \frac{3600 \mathrm{~s}}{1000 \, \mathrm{m}} = 54.72 \, \text{km/hr}\]

Step 2: Convert Volume from Liters to Gallons

There are approximately 0.264172 gallons in a liter. To convert the whale's lung capacity from liters to gallons, multiply the volume by this conversion factor:\[5.0 \times 10^{3} \, \text{L} \times 0.264172 \, \frac{\text{gallon}}{\text{L}} = 1320.86 \, \text{gallons}\]

Step 3: Convert Height from Feet to Meters

To convert height from feet to meters, use the conversion factor: 1 foot = 0.3048 meters. Multiply the height of the Statue of Liberty by this factor:\[151 \, \text{ft} \times 0.3048 \, \frac{\text{m}}{\text{ft}} = 46.02 \, \text{m}\]

Step 4: Convert Growth Rate from cm/day to inches/hour

First, convert centimeters to inches (1 cm = 0.393701 inches) and then handle the time conversion from days to hours (1 day = 24 hours). The calculation is as follows:\[60.0 \, \text{cm/day} \times 0.393701 \, \frac{\text{inch}}{\text{cm}} = 23.62206 \, \text{inch/day}\]Now, convert this to inches per hour:\[23.62206 \, \text{inch/day} \div 24 \, \text{hours/day} = 0.98425 \, \text{inch/hour}\]

Key Concepts

Speed Conversion

Speed conversion is all about changing the units of speed so the values mean the same thing, just in different dimensions. When converting the speed of a bumblebee flying at \( 15.2 \text{ m/s} \) to \( \text{km/hr} \), we apply specific conversion factors.

• 1 kilometer is 1000 meters, and
• 1 hour equals 3600 seconds.

To convert \( \text{m/s} \) to \( \text{km/hr} \), multiply the speed by 3600 and then divide by 1000:\[ 15.2 \text{ m/s} \times \frac{3600 \text{ s}}{1000} = 54.72 \text{ km/hr} \]This conversion factor effectively scales the speed to a more commonplace measure for everyday usage. Understanding the relation between these units is essential, especially when dealing with different types of data where standard units apply.

Volume Conversion

Volume conversion is transforming a volume measure from one unit to another. Let's focus on converting the blue whale's lung capacity from liters to gallons.
There is an approximate conversion factor:

• 1 liter is equivalent to 0.264172 gallons.

When using this conversion factor, we multiply the given volume in liters to get gallons:\[ 5.0 \times 10^{3} \text{ L} \times 0.264172 = 1320.86 \text{ gallons} \]Volume conversion is particularly useful in fields where different systems of measurement are used, like culinary arts or fuel estimation, ensuring consistency and comparability.

Height Conversion

Height conversion means translating height measurements from one unit of length to another. If we look at the example of the Statue of Liberty, converting feet to meters is straightforward.
The conversion factor is:

• 1 foot is equal to 0.3048 meters.

By multiplying the feet measurement by this factor, we get the height in meters:\[ 151 \text{ ft} \times 0.3048 = 46.02 \text{ m} \]Understanding height conversion is crucial in architecture and engineering when dimensions are often communicated in diverse unit systems across the globe. This ensures that structures meet local requirements without misunderstanding measurements.

Growth Rate Conversion

Growth rate conversion allows us to express how fast something is growing using different units for space and time. Here, the task was to convert bamboo's growth rate from centimeters per day to inches per hour.
Firstly, convert centimeters to inches:

• 1 cm equals 0.393701 inches.

Then multiply the growth rate:\[ 60.0 \text{ cm/day} \times 0.393701 = 23.62206 \text{ inches/day} \]To find the growth per hour, divide by the number of hours in a day:\[ 23.62206 \text{ inches/day} \div 24 = 0.98425 \text{ inches/hour} \]Adapting growth rates to different units can assist in understanding plant growth over various timeframes and environments, which can be essential in fields like agriculture and ecology.