class: center, middle, inverse, title-slide .title[ # Lesson 11: Problem-Solving with Units ] .author[ ### Fei Ye ] .date[ ### May, 2024 ] --- class: center middle <div> <style type="text/css">.xaringan-extra-logo { width: 110px; height: 25px; z-index: 0; background-image: url(assets/nc_sa.png); background-size: contain; background-repeat: no-repeat; position: absolute; bottom:12px;left:1%; } </style> <script>(function () { let tries = 0 function addLogo () { if (typeof slideshow === 'undefined') { tries += 1 if (tries < 10) { setTimeout(addLogo, 100) } } else { document.querySelectorAll('.remark-slide-content:not(.title-slide):not(.hide_logo)') .forEach(function (slide) { const logo = document.createElement('div') logo.classList = 'xaringan-extra-logo' logo.href = null slide.appendChild(logo) }) } } document.addEventListener('DOMContentLoaded', addLogo) })()</script> </div> ## Unit 2B: Problem-Solving with Units *Primary Source:* PPT for the book "Using & Understanding Mathematics". --- ## Unit Analysis in Problem Solving (1 of 2) - **Step 1.** Identify the units involved in the problem and the units that you expect for the answer. - **Step 2.** Use the given units and the expected answer units to help you find a strategy for solving the problem. Be sure to perform all operations (such as multiplication or division) on both the numbers and their associated units. --- ## Unit Analysis in Problem Solving (2 of 2) - Remember: - You cannot add or subtract numbers with different units, but you can combine different units through multiplication, division, or raising to powers. - It is easier to keep track of units if you replace division with multiplication by the reciprocal. For example, instead of dividing by 60 s/min, multiply by 1 min/60 s. - **Step 3.** When you complete your calculations, make sure that your answer has the units you expected. If it doesn't, then you've done something wrong. --- ## Example: Better deal You are planning to make pesto and need to buy basil. At the grocery store, you can buy small containers of basil priced at $2.99 for each 2/3-ounce container. At the farmer's market, you can buy basil in bunches for $12 per pound. Which is the better deal? **Solution:** To compare the prices, we need them both in the same units. Since 1 pound =16 ounces, the price for the small container of basil is $$ \dfrac{\$2.99}{\frac23\,\text{oz}}\times \frac{16\,\text{oz}}{1\, \text{lb}}\approx \$72/\text{lb} $$ The price of small containers is approximately six times as much as the farmer's market price. --- ## Example: When to stop for gas Your destination is 90 miles away, and your fuel gauge shows that your gas tank is one-quarter full. Your tank holds 12 gallons of gas, and your car averages about 25 miles per gallon. Do you need to stop for gas? **Solution:** To drive 90 miles with a fuel efficiency 25 miles per gallon, we need $$ \dfrac{90\,\text{mil}}{25\,\text{mil}/\text{gal}}=3.6\,\text{gal}. $$ One-quarter of a 12-gallon tank is only 3 gallons. Therefore, we should stop for gas. --- ## Units of Energy and Power **Energy** is what makes matter move or heat up. International metric unit is the **joule**. **Power** is the rate at which energy is used. International metric unit is the **watt**. $$ 1\, \text{watt}=\dfrac{1\, \text{joule}}{1\,\text{s}}. $$ A **kilowatt-hour** is a unit of energy. `$$1\, \text{kilowatt-hour} = 3.6\, \text{million joules}$$` --- ## Example: Operating Cost of a Light Bulb (1 of 2) A utility company charges 15 cents per kilowatt-hour of electricity. How much does it cost to keep a 100-watt light bulb on for a week? How much will you save in a year if you replace the bulb with an LED bulb that provides the same amount of light for only 25 watts of power? **Solution:** We firs find the energy consumed. $$ 100\,\text{W}\times\frac{1\,\text{kW}}{1000\,\text{W}}\times 1\,\text{wk}\times\frac{7\,\text{d}}{1\,\text{wk}}\times\frac{24\,\text{h}}{1\,\text{d}}=16.8 \text{kW-h}. $$ --- ## Example: Operating Cost of a Light Bulb (2 of 2) Now find the cost. $$ 16.8\,\text{kW-h}\times 15\frac{\text{cents}}{\text{kW-h}}=252\,\text{cents}=\$2.52. $$ The electricity for the bulb costs $2.52 per week. If you replace the 100-watt bulb with a 25-watt LED, you'll use only 1/4 as much energy, which means your weekly cost will be only 63 cents. In other words, your savings will be $2.52 - $0.63 = $1.89 per week, so in a year you'll save about: $$ \frac{\$1.89}{\text{wk}}\times\frac{52\,\text{wk}}{\text{yr}}\approx $98/\text{yr}. $$ --- ## Units of Density **Density** describes compactness or crowding. - *Material density* is given in units of mass per unit volume. e.g., grams per cubic centimeter ( `\(\text{g}/\text{cm}^3\)` ) - *Population density* is given by the number of people per unit area. e.g., people per square mile ( `\(\text{people}/\text{mi}^2\)` ) - *Information density* is given in units of mass per unit volume. e.g., gigabytes per square inch ( `\(\text{GB}/\text{in}^2\)` ) --- ## Units of Concentration **Concentration** describes the amount of one substance mixed with another. - The concentration of an air pollutant is often measured by the number of molecules of the pollutant per million molecules of air. e.g., parts per million (ppm) - Blood alcohol content (BAC) describes the concentration of alcohol in a person's body. e.g., grams of alcohol per 100 milliliters of blood --- ## Example: Medicine dosage (1 of 3) A child weighing 15 kilograms has a bacterial ear infection. A physician orders treatment with amoxicillin at a dosage based on 30 milligrams per kilogram of body weight per day, divided into doses every 12 hours. 1. How much amoxicillin should the child be prescribed every 12 hours? 2. If the medicine is to be taken in a liquid suspension with concentration 25 mg/ml, how much should the child take every 12 hours? --- ## Example: Medicine dosage (2 of 3) **Solution:** The prescribed dosage is 30 mg/kg of body weight per day, but because it will be given in two doses (every 12 hours), each dose will be based on half of the total, or 15 mg/kg of body weight. Therefore, for a child weighing 15 kilograms, the dosage should be $$ 15\,\text{kg}\times \frac{15\,\text{mg}}{\text{kg}}=225\,\text{mg}. $$ --- ## Example: Medicine dosage (3 of 3) **Solution: (Continued)** The liquid suspension contains 25 milligrams of amoxicillin per milliliter (ml) of liquid, and from part (a) we know the total amount of amoxicillin in each dose should be 225 mg. We are looking for the total amount of liquid that the child should be given for each dose, so the answer should have units of milliliters. $$ 225\,\text{mg}\div\frac{25\,\text{mg}}{\text{ml}}=225\,\text{mg}\times\frac{\text{ml}}{25\,\text{mg}}=9\,\text{ml}. $$ --- ## Practice: The silver value of one dollar .iframecontainer[ <iframe src="https://www.myopenmath.com/embedq2.php?id=587182&seed=2021&showansafter" width="100%" height="400px" data-external="1"></iframe> ] .footmark[ <a href="https://www.myopenmath.com/embedq2.php?id=587182&seed=2021&showansafter" target="_blank">Click here to open the practice in a new window</a> ] --- ## Practice: Price comparison A grocery sales the same egg in different ways. You can buy one dozen for $4.59 or 30 eggs for $10.99. Which way is cheaper? --- ## Practice: Tea or Chocolate .iframecontainer[ <iframe src="https://www.myopenmath.com/embedq2.php?id=271360&seed=2021&showansafter" width="100%" height="400px" data-external="1"></iframe> ] .footmark[ <a href="https://www.myopenmath.com/embedq2.php?id=271360&seed=2021&showansafter" target="_blank">Click here to open the practice in a new window</a> ] --- ## Practice: Utility bill .iframecontainer[ <iframe src="https://www.myopenmath.com/embedq2.php?id=677718&seed=2021&showansafter" width="100%" height="400px" data-external="1"></iframe> ] .footmark[ <a href="https://www.myopenmath.com/embedq2.php?id=677718&seed=2021&showansafter" target="_blank">Click here to open the practice in a new window</a> ] --- ## Practice: Ibuprofen Dosage .iframecontainer[ <iframe src="https://www.myopenmath.com/embedq2.php?id=587190&seed=2021&showansafter" width="100%" height="400px" data-external="1"></iframe> ] .footmark[ <a href="https://www.myopenmath.com/embedq2.php?id=587190&seed=2021&showansafter" target="_blank">Click here to open the practice in a new window</a> ] --- ## Practice: Unit Analysis on Blood alcohol content .iframecontainer[ <iframe src="https://www.myopenmath.com/embedq2.php?id=915593&seed=2023&showansafter" width="100%" height="400px" data-external="1"></iframe> ] .footmark[ <a href="https://www.myopenmath.com/embedq2.php?id=915593&seed=2023&showansafter" target="_blank">Click here to open the practice in a new window</a> ] --- ## Practice: Blood alcohol content A typical glass of wine contains about 20 grams of alcohol. Consider a 110-pound woman, with approximately 4 liters (4000 milliliters) of blood, who drinks two glasses of wine. 1. If all the alcohol were immediately absorbed into her bloodstream, what would her blood alcohol content be? 2. Again assume all the alcohol is absorbed immediately, but now assume her body eliminates the alcohol at a rate of 10 grams per hour. What is her blood alcohol content 3 hours after drinking the wine? Is it safe for her to drive at this time? Explain.