Problem 57 - Chemistry The Central Science In Si Units

Question

Perform the following conversions: (a) 5.00 days to s, (b) \(0.0550 \mathrm{mi}\) to \(\mathrm{m}\) (c) \(\$ 1.89 /\) gal to dollars per liter, (d) 0.510 in. \(/ \mathrm{ms}\) to \(\mathrm{km} / \mathrm{hr}\), (e) \(22.50 \mathrm{gal} / \mathrm{min}\) to \(\mathrm{L} / \mathrm{s}\) (f) \(0.02500 \mathrm{ft}^{3}\) to \(\mathrm{cm}^{3}\).

Step-by-Step Solution

Answer

5.00 days = 432,000 s; 0.0550 mi = 88.51 m; $1.89/gal = $0.4993/L; 0.510 in/ms = 46.63 km/hr; 22.50 gal/min = 1.4195 L/s; 0.02500 ft³ = 707.92 cm³.

Step 1: Convert Days to Seconds

To convert 5.00 days to seconds, we need to start by converting days to hours and then to minutes before converting to seconds. First, observe that there are 24 hours in a day. Thus, 5.00 days is converted as follows:\[5.00\, \text{days} \times \frac{24\, \text{hours}}{1\, \text{day}} = 120\, \text{hours}\]Next, convert hours to minutes. There are 60 minutes in an hour:\[120\, \text{hours} \times \frac{60\, \text{minutes}}{1\, \text{hour}} = 7200\, \text{minutes}\]Finally, convert minutes to seconds. There are 60 seconds in a minute:\[7200\, \text{minutes} \times \frac{60\, \text{seconds}}{1\, \text{minute}} = 432000\, \text{seconds}\]Thus, 5.00 days is equivalent to 432000 seconds.

Step 2: Convert Miles to Meters

To convert 0.0550 miles to meters, use the conversion factor that 1 mile is equal to 1,609.34 meters.\[0.0550\, \text{mi} \times \frac{1609.34\, \text{meters}}{1\, \text{mile}} = 88.5137\, \text{meters}\]Rounded to four significant figures, the conversion is 88.51 meters.

Step 3: Convert Dollars Per Gallon to Dollars Per Liter

To convert \\(1.89/\text{gal} to dollars per liter, note that 1 gallon is approximately 3.78541 liters:\[\frac{1.89\, \text{dollars}}{1\, \text{gallon}} \times \frac{1\, \text{gallon}}{3.78541\, \text{liters}} = \frac{1.89}{3.78541}\, \text{dollars/liter} \approx 0.4993\, \text{dollars/liter}\]Thus, \\)1.89 per gallon is approximately \$0.4993 per liter.

Step 4: Convert Inches Per Millisecond to Kilometers Per Hour

To convert 0.510 inches/ms to km/hr, start by converting inches to meters. One inch equals 0.0254 meters:\[0.510\, \text{in} \times \frac{0.0254\, \text{meters}}{1\, \text{inch}} = 0.012954\, \text{meters/ms}\]Since 1 ms is \(10^{-3}\) seconds, convert ms to seconds:\[0.012954\, \frac{m}{ms} \times 1000 = 12.954\, \frac{meters}{second}\]Convert meters/second to kilometers/hour using the fact that 1 km = 1000 m and 1 hour = 3600 seconds:\[12.954\, \text{m/s} \times \frac{1\, \text{km}}{1000\, \text{m}} \times \frac{3600\, \text{s}}{1\, \text{hr}} \approx 46.6344\, \text{km/hr}\]Thus, 0.510 inches/ms is approximately 46.63 km/hr.

Step 5: Convert Gallons Per Minute to Liters Per Second

To convert 22.50 gallons/minute to liters/second, use 1 gallon = 3.78541 liters:\[22.50\, \text{gallons/min} \times \frac{3.78541\, \text{liters}}{1\, \text{gallon}} = 85.171725\, \text{liters/min}\]Convert minutes to seconds by dividing by 60:\[\frac{85.171725\, \text{liters/min}}{60} \approx 1.4195\, \text{liters/second}\]Thus, 22.50 gallons/minute is approximately 1.4195 liters/second.

Step 6: Convert Cubic Feet to Cubic Centimeters

To convert 0.02500 cubic feet to cubic centimeters, note that 1 foot is 30.48 centimeters. Hence, 1 cubic foot is:\[(30.48\, \text{cm})^3 = 28316.846592\, \text{cm}^3\]Thus, convert the volume:\[0.02500\, \text{ft}^3 \times 28316.846592\, \text{cm}^3/ \text{ft}^3 = 707.9211648\, \text{cm}^3 \approx 707.92\, \text{cm}^3\]Thus, 0.02500 cubic feet is approximately 707.92 cubic centimeters.

Key Concepts

Days to Seconds Conversion

Time conversion might seem straightforward, but when breaking down the process from days to seconds, each step is crucial. Start with the number of days you want to convert, in this case, 5.00 days.

• First Step: Determine how many hours are in a day. Since there are 24 hours per day, multiplying 5.00 by 24 gives you 120 hours.
• Second Step: Next, convert hours into minutes. There are 60 minutes in an hour. Hence, multiply 120 hours by 60 to convert it to 7200 minutes.
• Final Step: Lastly, convert minutes to seconds. There are 60 seconds in a minute, so multiply 7200 minutes by 60 to get 432000 seconds.

So, 5.00 days equals 432000 seconds. This method ensures you understand the incremental conversions from units of large magnitude down to smaller units.

Miles to Meters Conversion

Converting distances from miles to meters is common in science and engineering. Here’s a simplified breakdown of the process using the conversion factor 1 mile = 1609.34 meters:

• Take your distance in miles, say 0.0550 miles.
• Multiply this value by the factor 1609.34 meters/mile to convert it to meters.

Performing the calculation, 0.0550 miles corresponds to approximately 88.51 meters when rounded to four significant figures. This is a key conversion for students and professionals working with international measurement standards.

Dollars per Gallon to Dollars per Liter

Fuel costs may initially be presented in USD per gallon within the United States. However, to understand costs elsewhere, conversion to USD per liter is necessary. Given the conversion 1 gallon = 3.78541 liters, here’s how the conversion works:

• Start with the cost in dollars per gallon, such as $1.89/gallon.
• Convert it using the ratio 1 gallon/3.78541 liters to find the cost per liter.

In this example, dividing $1.89 by 3.78541 gives approximately $0.4993 per liter.
This provides a more global perspective on pricing by aligning with internationally recognized units.

Inches per Millisecond to Kilometers per Hour

When dealing with high speeds, converting inches/millisecond to kilometers/hour is fascinating and practical. Below is a step-by-step guide to translating these units:

• Convert Inches to Meters: 1 inch equals 0.0254 meters. Multiply 0.510 inches by this factor to get meters per millisecond.
• Convert Milliseconds to Seconds: 1 millisecond equals 0.001 seconds, hence multiply the result by 1000 to get meters/second.
• Convert Meters/Second to Kilometers/Hour: Since 1 km = 1000 m and 1 hour = 3600 seconds, multiplying meters/second by (3600/1000) gives you kilometers/hour.

Performing these steps reveals that 0.510 inches/ms converts to about 46.63 km/hr. This conversion is useful in physics and engineering fields.

Gallons per Minute to Liters per Second

Flow rates involve converting gallons/minute to liters/second for comprehensive calculations, particularly in fluid dynamics.

• Start with the Flow Rate: Take 22.50 gallons per minute as the initial flow rate.
• Convert Gallons to Liters: Use the conversion 1 gallon = 3.78541 liters to find the flow rate in liters per minute.
• Convert Minutes to Seconds: Since 1 minute = 60 seconds, divide the liters/minute value by 60 to convert to liters/second.

This results in approximately 1.4195 liters per second, a common requirement in many industrial applications.

Cubic Feet to Cubic Centimeters

Volume conversion from cubic feet to cubic centimeters involves understanding linear conversions first. Here's how it works:

• Understand the Linear Conversion: Note that 1 foot = 30.48 centimeters.
• Calculate Cubic Conversion: Cube this value to find 1 cubic foot = 30.48³ cm³, which approximates to 28316.8 cm³.
• Apply to the Given Value: For 0.02500 cubic feet, multiply it by 28316.8 cm³/ft³.

This results in about 707.92 cm³. This conversion is often necessary in calculating volumes in various scientific fields.