Problem 45 - Chemistry The Central Science

Question

(a) A bumblebee flies with a ground speed of \(15.2 \mathrm{~m} / \mathrm{s}\). Calculate its speed in \(\mathrm{km} / \mathrm{h}\). (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) The bumblebee's speed is \(832\mathrm{~km/h}\). (b) The blue whale's lung capacity is \(1,320.86\mathrm{~gal}\). (c) The Statue of Liberty is \(46.0328\mathrm{~m}\) tall. (d) The bamboo's growth rate is \(59.0553\mathrm{~in/h}\).

Step 1: Write down the given speed in meters per second

The bumblebee's ground speed is given as \(15.2\mathrm{~m/s}\).

Step 2: Convert meters to kilometers

Since there are \(1,000\) meters in a kilometer, we'll divide the speed in meters by \(1,000\) to convert it to kilometers: \(\frac{15.2\mathrm{~m}}{1,000} = 0.0152\mathrm{~km}\).

Step 3: Convert seconds to hours

There are \(3600\) seconds in an hour, so we'll multiply the speed in seconds by \(3,600\) to convert it to hours: \(15.2\mathrm{~s} \times 3600 = 54,720\mathrm{~h^{-1}}\).

Step 4: Calculate the speed in kilometers per hour

Multiply the speed obtained in Step 2 by the factor obtained in Step 3: \(0.0152\mathrm{~km} \times 54,720 = 832\mathrm{~km/h}\). (b) Convert the blue whale's lung capacity from liters to gallons:

Step 1: Write down the given volume in liters

The blue whale's lung capacity is given as \(5.0 \times 10^{3}\mathrm{~L}\).

Step 2: Convert liters to gallons

There are approximately \(0.264172\) gallons in a liter, so we'll multiply the volume in liters by this factor to convert it to gallons: \(5.0 \times 10^{3}\mathrm{~L} \times 0.264172 = 1,320.86\mathrm{~gal}\). (c) Convert the height of the Statue of Liberty from feet to meters:

Step 1: Write down the given height in feet

The Statue of Liberty is \(151\mathrm{~ft}\) tall.

Step 2: Convert feet to meters

There are approximately \(0.3048\) meters in a foot, so we'll multiply the height in feet by this factor to convert it to meters: \(151\mathrm{~ft} \times 0.3048 = 46.0328\mathrm{~m}\). (d) Convert the bamboo's growth rate from centimeters per day to inches per hour:

Step 1: Write down the given growth rate in centimeters per day

The bamboo's growth rate is given as \(60.0\mathrm{~cm/day}\).

Step 2: Convert centimeters to inches

There are approximately \(0.393701\) inches in a centimeter, so we'll multiply the growth rate in centimeters by this factor to convert it to inches: \(60.0\mathrm{~cm} \times 0.393701 = 23.6221\mathrm{~in}\).

Step 3: Convert days to hours

There are \(24\) hours in a day, so we'll divide the growth rate in days by \(24\) to convert it to hours: \(\frac{60.0\mathrm{~cm}}{24} = 2.5\mathrm{~cm/h}\).

Step 4: Calculate the growth rate in inches per hour

Multiply the growth rate obtained in Step 2 by the factor obtained in Step 3: \(23.6221\mathrm{~in} \times 2.5 = 59.0553\mathrm{~in/h}\).

Key Concepts

Speed Calculation

Speed conversion is often necessary when dealing with different units like meters per second (\(\mathrm{m/s}\)) and kilometers per hour (\(\mathrm{km/h}\)). The bumblebee's speed example shows this process clearly.
To convert from \(\mathrm{m/s}\) to \(\mathrm{km/h}\), you need to know two things:

• 1 kilometer = 1000 meters
• 1 hour = 3600 seconds

First, convert meters to kilometers. If the bumblebee's speed is \(15.2\mathrm{~m/s}\), change meters into kilometers by dividing by 1000, which gives \(0.0152\mathrm{~km/s}\). Next, you convert seconds to hours by multiplying by 3600, resulting in \(54.72\mathrm{~km/h}\). Finally, to get the speed in \(\mathrm{km/h}\), multiply the converted figures: \(0.0152 \times 3600 = 54.72\).
This process ensures consistency in units, which is critical when interpreting speeds in different contexts.

Volume Conversion

Converting volume units is necessary for different scientific and real-world applications. For instance, converting the lung capacity of a blue whale from liters to gallons can help draw comparisons with more common experiences of volume.
To convert from liters (\(\mathrm{L}\)) to gallons (\(\mathrm{gal}\)), use the conversion factor:

• 1 liter = 0.264172 gallons

Given the blue whale lung capacity of \(5.0 \times 10^3\) liters, you multiply by \(0.264172\) to get \(1320.86\) gallons. Understanding these conversions can help translate abstract scientific measures into comprehensible terms for public discourse or further study.

Length Conversion

Converting lengths from feet to meters is critical in contexts like comparing building heights internationally. For example, the Statue of Liberty is initially measured in imperial units (feet), which might not be immediately understandable for everyone worldwide.
For conversion:

• 1 foot = 0.3048 meters

Take the Statue's height at 151 feet and multiply by \(0.3048\) to result in a height of approximately \(46.0328\) meters. This conversion is valuable as it allows easier conceptualization of heights in the metric system, which is globally prevalent.

Growth Rate Conversion

Converting growth rates is particularly useful in botany and agriculture when comparing plants' growth under varying conditions. This might be the case when converting bamboo's growth from \(\mathrm{cm/day}\) to inches per hour (\(\mathrm{in/h}\)).
First, convert centimeters to inches using:

• 1 centimeter \(\approx 0.393701\) inches

The initial growth of 60 \(\mathrm{cm/day}\) translates to \(23.6221\) inches. To change days to hours, remember:

• 1 day = 24 hours

So, \(60\mathrm{~cm} \) equates to about \(2.5\mathrm{~cm/hour}\). Final multiplication gives us the rate in inches per hour: \(23.6221 \times 2.5 = 59.0553\) inches per hour. This procedure allows researchers and agriculturalists to adapt plant growth figures to various imperial or metric system needs.