4.6: Corequisite- Dimensional Analysis, Unit Conversion

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SPECIFIC OBJECTIVES

By the end of this lesson, 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.
formulas can be found by searching the Internet and reference books.
a variable can be used to represent an unknown.
using a formula requires knowing what each variable represents.

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

calculate quantities in the billions.
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.
use formulas from geometry and perform calculations that involve rates and measures to support financial decisions.
evaluate an expression.
use the appropriate units for length, area, and volume.

PROBLEM SITUATION 1: 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.¹³ 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? 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? Let’s work through answering this question using dimensional analysis.

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

$Cents over minutes.$

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

$100 cents over 1 dollar is cents over minutes.$

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:

$100 cents over 1 dollar times 1200 dollars over two weeks. The dollar signs cancel.$

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

$100 cents over 1 dollar times 1200 dollars over two weeks. times 1 week over 5 days. The weeks cancel.$

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

$100 cents over 1 dollar times 1200 dollars over two weeks times 1 week over 5 days times 1 day over 8 hours times 1 hour over 60 minutes. The dollars, weeks, days, and hours cancel.$

(2) Is the resulting calculation reasonable? Explain.

(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: Remember that there are 5,280 ft in one mile. 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.

(4) Let's examine population densities and use these to calculate projected populations. The population density of Tokyo is 6,038 people per square kilometer (km). Use dimensional analysis to calculate how many people would live in the nation of Japan, which comprises an area of approximately 378,000 square km, if the entire nation was as dense as the city of Tokyo.

(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. Use the same procedure as in the previous question.

PROBLEM SITUATION 2: HOME IMPROVEMENTS

Bob and Carol Mazursky have purchased a house and the small piece of land that surrounds the house. A small piece of land that can be purchased or sold is often called a lot. Bob and Carol want to make some improvements to their new property. In the following few problems, you will calculate the costs of these improvements. Scale drawings of the house and the lot are displayed in the problems. You will need these drawings to calculate improvement costs.

Figure 1

$Scale drawing showing the House and lot, indicating the following scale: 1 square = 10 feet 2 squares = 20 feet 3 squares = 30 feet$

(7) Review the drawing of the house and lot (Figure 1). What does the scale mean?

(8) Bob and Carol want to get the backyard (Figure 1) fertilized and reseeded. They found an ad for Gerry’s Green Team lawn service, which can be seen here. Gerry came to their house and said that the job would take about half a day and would cost about $600. Is Gerry’s estimate consistent with his advertisement? Why or why not?

$Gerry's Green Team flyer. Grass seed = 4 pounds per 1000 sg ft at !1.25 per pound. Fertilizerr: 50 pounds per 12000 sq ft at $0.50 per pound. Labor: 4 hours at $45 per hour.$

(9) There is a brick grill in the backyard. Bob and Carol are going to make a concrete patio in the shape of a trapezoid next to the grill (see below). The concrete slab needs to be at least 2 inches thick. They will use 40-pound bags of premixed concrete. Each 40-pound bag makes 0.30 cubic feet of concrete and costs $6.50. How much will the materials cost, including the 7.5% tax?

Outdoor Grill: Bob and Carol are going to add a trapezoidal patio adjacent to the outdoor grill in the backyard. The shaded area is made of concrete and two inches deep.

$Scale image of grill and patio design. Grill is rectangular with length of 6 feet and height of 3 feet. Grill is a trapezoid with length 6 feet and height of 3 feet.$

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¹³ https://www.toyota.com/prius/

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