Problem 47 - Chemistry: The Central Science

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 \(151 \mathrm{ft}\) 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

The short answers are: (a) The bumblebee's speed is \(54.72 \mathrm{~km/hr}\), (b) the blue whale's lung capacity is \(1320.86 \mathrm{~gallons}\), (c) the Statue of Liberty is \(46.0224 \mathrm{~meters}\) tall, and (d) bamboo grows at a rate of \(0.3937 \mathrm{~inches/hr}\).

Step 1: Conversion of speed from m/s to km/hr

Given that the bumblebee flies with a speed of 15.2 m/s, we need to convert this to km/hr. The conversion factor is 3.6 (i.e., 1 m/s is equivalent to 3.6 km/hr). This can be done by using the formula: \[ \text{{Speed in km/hr}} = \text{{Speed in m/s}} \times 3.6 \] Substituting the given speed into the formula: \[ \text{{Speed in km/hr}} = 15.2 \times 3.6 = 54.72 \, \text{{km/hr}} \]

Step 2: Conversion of volume from L to gallons

To convert the volume of L to gallons, we use the conversion factor 0.264172 (i.e., 1 L is equivalent to 0.264172 gallons). Hence, \[ \text{{Volume in gallons}} = \text{{Volume in L}} \times 0.264172 \] Substituting the given volume into the formula: \[ \text{{Volume in gallons}} = 5.0 \times 10^{3} \times 0.264172 = 1320.86 \, \text{{gallons}} \]

Step 3: Conversion of height from ft to meters

To convert the height from feet to meters, we use the conversion factor 0.3048 (i.e., 1 ft is equivalent to 0.3048 meters). Hence, \[ \text{{Height in meters}} = \text{{Height in ft}} \times 0.3048 \] Substituting the given height into the formula: \[ \text{{Height in meters}} = 151 \times 0.3048 = 46.0224 \, \text{{meters}} \]

Step 4: Conversion of growth rate from cm/day to inches/hr

Given the rate of growth as 60 cm/day, we need to convert this to inches/hr. First, let's convert cm to inches using the conversion factor that 1 inch equals to 2.54 cm. Then, convert days to hours, so 1 day = 24 hours. Hence, \[ \text{{Growth rate in inches/hr}} = \left(\frac{{\text{{Growth rate in cm/day}}}}{{2.54}}\right) \times \frac{1}{24} \] Substituting the given growth rate into the formula: \[ \text{{Growth rate in inches/hr}} = \left(\frac{60}{{2.54}}\right) \times \frac{1}{24} = 0.3937 \, \text{{inches/hr}} \]

Key Concepts

Converting Speed Units

Understanding how to convert speed units, such as from meters per second (m/s) to kilometers per hour (km/hr), is essential in physics and everyday life, especially when the contexts require different units. For example, the speed of a bumblebee or a car may be expressed in different units depending on the geographic location or scientific study.

When converting speed, keep in mind that one unit of speed measures how much distance is covered within a certain time frame. The key is to align the units of distance and time across both measurements. To convert from meters per second to kilometers per hour, multiply the speed by the conversion factor of 3.6. This is because 1 kilometer is 1000 meters, and 1 hour is 3600 seconds, so the conversion factor simplifies to 3.6.

Volume Conversion

Volume conversion is a frequent task in chemistry and cooking among other fields. Different regions and disciplines use various units to measure volume, which includes liters (L) and gallons. When you encounter a situation where you need to convert these units, it's crucial to know the proper conversion factor.

To convert liters to gallons, one can use the conversion factor of 0.264172, since a single liter is equivalent to approximately 0.264172 gallons. This conversion ratio is foundational when dealing with liquids in different systems of measurement, such as converting the large volume of a whale's lung capacity to a more familiar unit in the U.S.

Length Measurement Conversion

Converting units of length, such as from feet (ft) to meters (m), is a fundamental skill in a number of fields from construction to international travel. Different parts of the world utilize different systems of measurement; the U.S. commonly uses feet while most other countries use meters.

To carry out this conversion, the factor of 0.3048 is used because one foot equals 0.3048 meters. It's helpful to remember this factor when traveling or working on international projects. For example, the height of famous landmarks like the Statue of Liberty is often recorded in feet, but converting it to meters allows for a universal understanding of its stature.

Growth Rate Calculation

Growth rate calculation can be applied to study the rate at which plants, populations, or investments grow over time. In botany, for instance, understanding how fast a plant or a tree like bamboo grows, can be crucial for ecological studies or agriculture. Growth can be quantified in terms of length per time, such as centimeters per day, and might need to be converted to other units like inches per hour depending on the requirements.

The process of conversion involves two steps: changing the unit of length, typically using the conversion of 2.54 centimeters per inch, and then altering the time unit from days to hours, knowing that a day consists of 24 hours. The ability to perform these conversions accurately ensures correct and meaningful comparisons across different studies or applications.