Alg 2 BC - Unit 2Day 9 - Linear Models

Linear Modeling:

Lines are often useful in modeling real-life situations and making predictions.

Example: Jacksonville, Florida has an elevation of 12 ft above sea level. A hot air balloon taking off from Jacksonville rises 50ft/min.

a) Write an equation to model the balloon’s elevation as a function of time.

b) Graph the equation

c) Interpret what the y-intercept means in terms of the problem.

Example: A candle is 6 inches tall after burning for 1 hour. After 3 hours, it is 5.5 inches tall.

a) Write an equation to model the candle’s height as a function of time.

b) What does the slope mean in terms of the problem?

c) What does the y-intercept mean in terms of the problem?

d) How tall will the candle be after burning for 11 hours?

e) When will the candle burn out?

Try One:

As you drive home from the football game, the number of miles you are from home depends on the number of minutes you have been driving. Suppose that you are 11 miles from home when you have been driving for 10 minutes and 8 miles from home when you have been driving for 15 minutes.

a.Write an equation to model distance as a function of time.

b.Predict your distance from home when you have been driving for 20 minutes.

c.When were you 7 miles from home?

d.Calculate the x-intercept. (let y = 0 and solve for x) What does this number tell you in the context of the problem?

e.Calculate the y-intercept. (let x = 0 and solve for y) What does this number tell you in the context of the problem?

f. What does the slope tell you?

1. Scientists have found that the rate at which crickets chirp is a linear function of temperature. At 59°F they make 76 chirps per minute, and at 65°F they make 100 chirps per minute.

a.Write the linear equation expressing chirping rate in terms of temperature.

b.Predict the chirping rate for 90°F.

c.How warm is it if you count 120 chirps per minute?

d.Calculate the temperature-intercept. What does this number tell you?

e.Calculate the chirping-rate-intercept. What does this number tell you?

f.Sketch the graph of this function in a reasonable domain.

2. Suppose you own a car that is presently 40 months old. After doing some research, you find that its present value trade-in value is $3300. You also find that the trade in value 10 months ago was $4700.

a.Write a linear equation expressing the car’s trade-in value as a function of the car’s age in months.

b.You plan to get rid of the car when its trade-in value is $1000. How much longer can you keep the car?

c.By how many dollars does the car depreciate (decrease in value) each month? What part of your equation tells you this?

d.When do you predict that the car will be worthless? What part of the equation tells you this?

e.According to your linear model, what was the trade-in value when your car was new?

3. Bridges on expressways often have expansion joints, which are small gaps in the roadway between one section of the bridge and another. The gaps are put there so that the bridge will have room to expand when the weather gets hot. Suppose that a bridge has a gap of 1.3 cm when the temperature is 22°C and that the gap narrows to 0.9 cm when the temperature warms to 30°C.

a.Write a linear equation that expresses gap width in centimeters as a function of temperature in degrees Celsius.

b.How wide would the gap be at 35°C? At -10°C?

c.At what temperature would the gap close completely?

d. Interpret the slope in terms of cm and temperature.

4. Andy sells one-gallon cartons of milk (4 quarts) for $3.09 each and half-gallon cartons for $1.65 each. Assume that the number of cents you pay for a carton of milk varies linearly with the number of quarts the carton holds.

a.Write the linear equation that expresses price in cents as a function of the number of quarts in the carton.

b.If Andy sold 3 gallon cartons, what would the equation predict the price to be?

c.The actual prices for pint cartons (1/2 quart) and one-quart cartons are $0.57 and $0.99 respectively. Do these prices fit your equation? If not, are they higher or lower than predicted?

d.Suppose that you found cartons of milk marked at $3.45, but there was nothing on the carton to tell what size it is. According to you equation, how much would this carton hold?

e. What does the y-intercept represent in the real-world?

f..Interpret the slope in terms of price and number of quarts.