Section MU.4 – Problem Solving Using Rates and Dimensional Analysis
Many problems can also be solved by multiplying a quantity by rates to change the units. This is the foundation of the Factor-Label process that we have been using already in this chapter.
| Example 1 |
| The winner of the Tour de France has posted an overall average speed of 25 miles per hour (mph). How fast was the bicyclist going in kilometers per hour (kmh)? |
| We will use 1 mile = 1.61 kilometers.
[latex]\frac{25~miles}{1~hour}[/latex] • [latex]\frac{1.61~kilometers}{1~mile}[/latex] = ___ kph [latex]\frac{25~\bcancel{miles}}{1~hour}[/latex] • [latex]\frac{1.61~kilometers}{1~\bcancel{mile}}[/latex] = ___ kph [latex]\frac{25~\cdot~1.61~kilometers}{1~hour~\cdot~1}[/latex] = [latex]\frac{40.25~kilometers}{1~hour}[/latex] The winner of the Tour de France averaged 40.25 kilometers per hour. |
| You Try MU.4.A |
| A commercial plane traveled at 880 kilometers per hour. What is this speed in miles per hour? Round to the nearest tenth. |
| Example 2 |
| A rectangular pool requires 168 feet of fencing. If fences cost $0.35 per inch, how much will it cost to enclose the pool area? |
| We first need to convert feet into inches.
[latex]\frac{168~\bcancel{feet}}{1}[/latex] • [latex]\frac{12~inches}{1~\bcancel{foot}}[/latex] = 2,016 inches Now we can find the total cost of the fencing. [latex]\frac{2,016~\bcancel{inches}}{1}[/latex] • [latex]\frac{$0.35}{1~\bcancel{inch}}[/latex] = $705.60 The cost to enclose the pool area is $705.60. |
| Example 3 |
| Susan would like to wallpaper her room. She calculates that she will need 400 square feet of wallpaper. Each roll of wallpaper contains 4 square feet and costs $30.99. How much will Susan spend on wallpaper? |
| Since wallpaper rolls are sold per square yard, we need to convert square feet into square yards.
[latex]\frac{400~\bcancel{ft^2}}{1}[/latex] • [latex]\frac{1~yd~\cdot~1~yd}{3~\bcancel{ft}~\cdot~3~\bcancel{ft}}[/latex] = 44.44 yd2 Now we will calculate how many rolls we will need. [latex]\frac{44.44~\bcancel{yd^2}}{1}[/latex] • [latex]\frac{1~roll}{4~\bcancel{yd^2}}[/latex] = 11.11 rolls Susan will need to buy 12 rolls of wallpaper. [latex]\frac{12~\bcancel{rolls}}{1}[/latex] • [latex]\frac{$30.99}{1~\bcancel{roll}}[/latex] = $371.88 Susan will spend $371.88 on wallpaper. |
| You Try MU.4.B |
| Susan would like to paint the other three bedrooms in the house. She determines that she has 200 square yards to paint. The paint store sells gallons for $15.99. Each gallon covers approximately 350 square feet. How much will it cost her to paint the other three rooms? |
| Example 4 |
| A bicycle is traveling at 15 miles per hour. How many feet will it cover in 20 seconds? |
| To answer this question, we need to convert 20 seconds into part of an hour. If we know the speed of the bicycle in feet per second, this question would be simpler. Since we don’t, we will need to do additional unit conversions. We will need to know that 5,280 ft = 1 mile. We might start by converting the 20 seconds into hours:
[latex]\frac{20~\bcancel{seconds}}{1}[/latex] • [latex]\frac{1~\bcancel{minute}}{60~\bcancel{seconds}}[/latex] • [latex]\frac{1~hour}{60~\bcancel{minutes}}[/latex] = [latex]\frac{1}{180}[/latex] hour Now we can multiply by the 15 milers per hour. [latex]\frac{1~\bcancel{hour}}{180}[/latex] • [latex]\frac{15~miles}{1~\bcancel{hour}}[/latex] =[latex]\frac{1}{12}[/latex] mile Now we can convert to feet. [latex]\frac{1~\bcancel{mile}}{12}[/latex] • [latex]\frac{5,280~feet}{1~\bcancel{mile}}[/latex] = 440 feet We could have also done this entire calculation in one long set of products: [latex]\frac{20~\bcancel{seconds}}{1}[/latex] • [latex]\frac{1~\bcancel{minute}}{60~\bcancel{seconds}}[/latex] • [latex]\frac{1~\bcancel{hour}}{60~\bcancel{minutes}}[/latex] • [latex]\frac{15~\bcancel{miles}}{1~\bcancel{hour}}[/latex] • [latex]\frac{5,280~feet}{1~\bcancel{mile}}[/latex] = 440 feet |
| You Try MU.4.C |
| A car is traveling at 80 kilometers per hour. How many meters does it travel in 10 seconds? Round to the nearest tenth. |
| Example 5 | ||||||||||||||
You are walking through a hardware store and notice two sales on tubing:
Either tubing is acceptable for your project, which tubing is the least expensive? |
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Find the unit price for each tubing. This will make it easier to compare.
The cost of tubing A is $1.83 per yard.
The cost of tubing B is $0.94 per foot. To compare the prices, you need to have the same unit of measure. You can choose to use dollars per yard (like we have for Tubing A) or dollars per foot (like we have for Tubing B), either will work. For this example, we will go with dollars per yard.
Compare prices for 1 yard of each tubing. Tubing A: $1.83 Tubing B: $2.82 Tubing A is the least expensive. |
| Example 6 |
| The cost of gasoline in Arizona is about $2.05 per gallon. When you travel over the border into Mexico, gasoline costs 14.81 pesos per liter. Where is gasoline more expensive?
Note: This problem requires a currency conversion factor. Currency conversions are constantly changing, for this example $1 = 19.67 pesos. |
| To answer this question, we need to convert from gallons to liters AND from U.S. dollars to Mexican pesos.
[latex]\frac{$~2.05}{1~gallon}[/latex] • [latex]\frac{1~gallon}{3.79~liters}[/latex] • [latex]\frac{19.67~pesos}{$~1}[/latex] = ___ pesos per liter [latex]\frac{\bcancel{$}~2.05}{1~\bcancel{gallon}}[/latex] • [latex]\frac{1~\bcancel{gallon}}{3.79~liters}[/latex] • [latex]\frac{19.67~pesos}{\bcancel{$}~1}[/latex] = ___ pesos per liter [latex]\frac{2.05~\cdot~19.67~pesos}{3.79~liters}[/latex] = 10.64 pesos per liter The price in Arizona of $2.05 per gallon is equivalent to 10.64 pesos per liter. Since the actual price in Mexico is 14.81 pesos per liter, gasoline is more expensive in Mexico. |
Section MU.4 – Answers to You Try Problems
MU.4.A
546.6 miles per hour (mph)
MU.4.B
$95.94
MU.4.C
222.2 meters
