Engineering Applications Unit 3 Solving Problems that involve Linear Equations 0-2 weeks

Standard-Engineering Apps. EA.2.2. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. EA.2.4. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. EA.2.6. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

Standard-Engineering Apps. EA.3.1. Distinguish between situations that can be modeled with linear functions and with other functions including recognizing situations in which one quantity changes at a constant rate per unit interval relative to another. EA.3.2. Interpret the parameters in linear and other function types in terms of a context. EA.5.1 Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.

Standard-Engineering Apps. EA.5.2 Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and trigonometric models. EA.5.3 Informally assess the fit of a function by plotting and analyzing residuals. EA.5.4 Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. EA.5.5 Compute (using technology) and interpret the correlation coefficient of a linear fit.

Prerequisites Using signed numbers and vectors Using scientific notation Solving problems with powers and roots Using Formulas to solve problems

Remember An equation is a mathematical sentence that says one quantity is equal to another quantity. Variable is a letter that represents the an unknown quantity. You may choose a letter that reminds you of this quantity. Slope-intercept form ◦ y = mx + b Slope Y-intercept

Linear Equations Variable(s) in linear equations ◦ Cannot have exponents ◦ Cannot multiply or divide each other ◦ Cannot be found under a root sign or square root sign

Translate words to equations Read and Understand the problem Assign a variable to the quantity you are trying to find Write what the variable represents Re-read the problem and write an equation Solve the equation Answer the question in the problem Check your solution

Example 1 A can of soft drink holds twelve ounces. How many ounces are in a case of 24 ounce cans? V =24 cans 12 oz per can V = V = 288 ounces The volume of a soft drink in a case of 24 cans is equal to the number of cans times the volume of one can

Example 2 A metal pipe is 20.8 inches long. How much must be cut off to leave a piece that 12.5 inches long. p = 20.8 in – 12.5 in p = 20.8 – 12.5 p = 8.3 inches The piece of pipe to be cut off is equal to the total length of the pipe minus the amount of pipe that is to remain.

Example 3 The family budget has ten percent of the take-home pay assigned to transportation. If the monthly take-home pay is $1200, how much is in the budget for transportation. t = 10% $1200 t = t = $120 The amount budgeted for transportation is equal to the total take-home pay multiplied by the percent that is assigned to transportation.

Solve 3g – 7 = = +7 3g = 18 3 = 3 g = 6

xy Equation: y = 3x

Example 4 You’ve just been hired as a salesclerk in a large department store. You’ve been told that you will receive 5% of the total value of the sales you make during the week. How can you predict your gross pay for any week?

Example 4 Your weekly salary is $150 plus 5% of your total sales for the week. Let w = weekly pay Let s = sales

Example 4 Solve ◦ w = 0.05s w = $325

Example 5 An apartment manager is concerned about the weight of a waterbed a tenant wants to bring in. A king-size waterbed holds about 150 gallons of water. A gallon of water weighs approximately 8.3 pounds. The water bed frame weighs close to 80 pounds. How much will the filled waterbed, frame and all, weigh?

Example 5 Let w = total weight of water and frame Let v = volume of water (in gal) Let f = weight of bed frame alone

Example 5 Weight = (8.3)(Volume of water) + Frame w = 8.3v + f w = 8.3(150) + 80 w = 1325 lb

Challenge A racing car passes a “start line” traveling at a speed of 44 ft/sec (same as 30 mph). It increases its speed (accelerates) at a rate of 17 ft/sec during each second of travel. Write a formula that gives the car’s speed in V (in ft/sec) for any time t (in seconds) after it passes the “start line”

Challenge V = 17t + 44 Use your formula to calculate the car’s speed (in ft/sec) 10 seconds after it passes the “start-line”. t = 10 seconds V = V = 214 ft/sec

Challenge Express the speed in ft/sec as an equivalent speed in miles/hour.