3.1: Salary per Minute

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INTRODUCTION

In 1998, “NASA lost its $125-million Mars Climate Orbiter because spacecraft engineers failed to convert from English to metric measurements when exchanging vital data before the craft was launched. A navigation team at the Jet Propulsion Laboratory used the metric system of millimeters and meters in its calculations, while Lockheed Martin Astronautics in Denver, which designed and built the spacecraft, provided crucial acceleration data in the English system of inches, feet, and pounds. As a result, JPL engineers mistook acceleration readings measured in English units of pound-seconds for a metric measure of force called newton-seconds.”

"People make mistakes all the time," said Carl Pilcher, the agency's science director for solar system exploration. "I think the problem was that our systems designed to recognize and correct human error failed us.”

Even though this was an expensive mistake ($125 million!), the mistake allowed the engineers to recognize that their system for catching errors (like conversion mistakes) was not good enough and had to be improved. Not all mistakes cost $125 million, but mistakes often provide an opportunity to make something better!

In this collaboration, we will be discussing how to convert units both between and within the metric and the US measurement systems. The procedure is called dimensional analysis, or the Unit Factor Method. Dimensional analysis is commonly used by healthcare professionals, computer scientists, or NASA engineers, just to mention a few professions.

Quick Check-in:

Please discuss the following questions with your group:

What specific concept from Unit 2.9 do you feel fairly confident about?
Are there any specific concepts from Unit 2.9 that you find challenging?

SPECIFIC OBJECTIVES

By the end of this collaboration, you should understand that

the units found in a solution may be used as a guide to the operations required in the problem—that is, factors are positioned so that the appropriate units cancel.
units can add meaning to the numbers that result from calculations.

By the end of this collaboration, you should be able to

write a rate as a fraction.
use a unit factor to simplify a rate.
use dimensional analysis to help determine the factors in a series of operations to obtain an equivalent measure.

PROBLEM SITUATION: USING DIMENSIONAL ANALYSIS

Dimensional analysis is a method of setting up problems that involves converting between different units of measurement. It is also called unit analysis or unit conversion. Many professionals—including pharmacists, dieticians, lab technicians, and nurses—use dimensional analysis. It is also useful for everyday conversions in cooking, finances, and currency exchanges. Many people can do simple conversions without dimensional analysis; however, they will likely make mistakes on more complex problems.

The advantage of using dimensional analysis is that it is a way to check your calculations. While it is always important that you develop your own methods to solve problems, this is a time when you are encouraged to learn and use a specific method. Once you have learned dimensional analysis, you can decide when to use it and when to use other methods.

(1) (a) According to Toyota’s website, a 2023 Prius can get an estimated 57 miles per gallon (mpg) in the city and 56 mpg on the highway.¹ How many miles will you be able to drive in the city if you have 4.5 gallons of gas?

(b) How many gallons of gas will you need to drive 3,450 miles on the highway? Round your answer to one decimal place.

Converting Your Paycheck

This is an example of how to use dimensional analysis to solve a problem.

Sample Question: Your paycheck for two weeks came out to $1,200. You work eight hours a day, five days a week. How much are you making per minute in cents? (Hint: You will need to use dimensional analysis to solve this problem.)

Answer: First, decide what you are looking for. Here you are looking for the rate of cents per minute, which can be written as:

\[\dfrac{c}{min}\nonumber \]

Now, identify a rate that we know has the same numerator )the part of the fraction that represents a count of the number of parts) as the rate we are looking for. Since cents converts to dollars, we have:

\[\dfrac{100c}{$1} \rightarrow \dfrac{c}{min}\nonumber \]

Next, we want to cancel the “$” (since this unit is not in the answer we are looking for). To cancel the “$” we can multiply by a unit ratio with “$” in the numerator. Since $1,200 is the amount we make in two weeks, we can multiply by that ratio:

\[\dfrac{100c}{$1} \times \dfrac{$1200}{2\;weeks} \rightarrow \dfrac{c}{min}\nonumber \]

Now, we need to cancel the unit “weeks”, so our next ratio must have “weeks” in the numerator:

100 cents over $1 times $1200 over 2 weeks times week over 5 days goes to cents over minutes.

Continuing this process, we finally get to the unit factor for “minute”:

100 cents over $1 times $1200 over 2 weeks times week over 5 days times day over 8 hours times hour over 60 minutes goes to cents over minutes.

(2) Is the resulting calculation reasonable? Discuss with your group.

(3) Many states have banned texting while driving because it is dangerous, but many people do not think that texting for a few seconds is that harmful. Suppose you are driving 60 miles/hour and you take your eyes off the road for four seconds. How many feet (ft) will you travel in that time? Hint: Start with the unit you are looking for (ft). Then, create a chain of ratios, starting with one where “ft” is in the numerator that will cancel all other units. Remember that there are 5,280 ft in one mile.)

(4) In Module 2 we examined population densities and used these to calculate projected populations. The population density of Tokyo² is 6,158 people per square kilometer (km). Use dimensional analysis to calculate how many people would live in the nation of Japan³, which comprises an area of approximately 378,000 square km, if the entire nation was as dense as the city of Tokyo.

Dimensional analysis is an essential skill in dosage calculations, which are necessary for any nurse to master. The following two questions will be more difficult to answer without using dimensional analysis.

(5) Nurses are often required to calculate dosages. That is, they must check the order that a doctor has given for the administration of a drug and decide whether the dosage is correct. To calculate correctly they must convert between different metric units. For example, 1,000 milligrams (mg) = 1 gram (g); and 1,000 micrograms (mcg) = 1 mg.

Suppose a doctor has ordered a dose of 0.1 gram of a medication. The drug comes in a solution concentration of 200 mg per milliliter. How many milliliters of this solution is required?

(6) Now, calculate how many milliliters you would need to administer 500 mg from a dosage concentration of 1 g per 3 mL.

FURTHER APPLICATIONS

(7) Do an Internet search for “dimensional analysis” or “unit analysis” or “unit factor method.” Find at least one site that provides examples of how to make conversions using this technique.

(a) Record the site name and URL address.

(b) Copy one example of a conversion using dimensional analysis (as shown on the site).

MAKING CONNECTIONS

Record the important mathematical ideas from the discussion.

__________________________________

¹ https://www.toyota.com/prius/

² https://www.metro.tokyo.lg.jp/ENGLISH/ABOUT/HISTORY/history03.htm

³ https://www.cia.gov/the-world-factbook/countries/japan/#geography

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