Question:

Linear Models

1. The value of a machine depreciates by an equal amount each year of its useful

life. This "constant change" allows us to model this situation using a linear equation. Suppose the

machine was purchased for $75,600 and has a useful life of seven years, after which time it has a value

of $0.

(a) Identify the x-variable and the y-variable.

(b) Construct the data table for this problem.

(c) Find the equation of the linear model.

(d) What is the annual depreciation rate?

(e) What is the value of the machine five years after its purchase?

(f) When will the value of the machine be $45,000?

Give your final answer in terms of years and months.

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2. Units Produced Total Cost

1,000 $ 93,250

2,000 $ 117,100

3,000 $ 140,950

4,000 $ 164,800

(a) Identify the x-variable and the y-variable.

(b) Use the definition of linearity to convince yourself that this data set is exactly linear.

(c) Find the equation of the linear model.

(d) Identify an data analysis table: ( x , data: y , model: y )

Use this table to assess the fit of the model to the data.

(e) Find the equation for C(x).

(f) What is the fixed cost and variable cost for this production?

(g) Use this model to estimate the total cost of producing 5,000 units.

(h) How many units can be produced if you have a total production budget

of $200,000?

The Data Analysis Table shows us that the equation really does describe the data set, and in the case of

exactly linear data, describes it perfectly.

3. You are the proud owner and operator of the mall franchise for the world famous

"Crazy Uncle Jack's Cajun Catfish Shack", home of the delicious catfish hot dog.

You have some flexibility in determining the selling price of the hot dogs.

Consider the following data, collected on "similar" days at the mall:

Price of hot dog # of hot dogs sold

$ 1.50 70

$ 2.50 48

$ 3.00 37

$ 4.00 15

(a) Identify the x-variable and the y-variable.

(b) Use the definition of linearity to convince yourself that this data set is exactly linear.

(c) Find the equation of the linear model.

(d) Identify a data analysis table: ( x , data: y , model: y )

Use this table to assess the fit of the model to the data.

(e) What should the selling price be in order to sell approximately 25 hot dogs?

(f) Approximately how many hot dogs will you expect to sell if the selling price

is $2.00?

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4. The value of a machine depreciates by an equal amount each year of its useful

life. Suppose the machine was purchased for $18,000 and has a useful life of five

years, after which time it has a scrap value of $2,000.

(a) Identify the x-variable and the y-variable.

(b) Construct the data table for this problem.

(c) Find the equation of the linear model.

(d) What is the annual depreciation rate?

(e) What is the value of the machine two and one-half years after its purchase?

(f) When will the value of the machine be $6,000?

Give your final answer in terms of years and months.

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5. There are two common units of measurement for temperature: degrees Fahrenheit (F) and degrees

Celsius (C).

Water freezes at: 0 C and 32 F

Water boils at: 100 C and 212 F

Can we derive a linear formula by converting from F to C and back again?

Of course we can!

(a) Use the above data to find a formula for F as a function of C

Hint: leave m as a fraction

Hint Hint: F as a function of C is the same thing as x as a function of y, so here, your F is your x and

C is your y.

(b) Use the above data to find a formula for C as a function of F

Hint: leave m and b as functions

(c) Use the formula from (a) and algebraic manipulation to find a formula for C as a function of F.

Compare your answer from (b)

(d) You are going on a business trip to Bzwakii, where the average temperature is 11 C. How should you dress?

(e) You are sending a recipe for Butterscotch Brownies to your friend in Canada. Your recipe calls for the brownies to be baked at 350 F, What is the equivalent temp in C?