Question

assume the following situation can be modeled by a linear function. write an equation for the linear function and use it to answer the given question. be sure you clearly identify the independent and dependent variables. then briefly discuss whether a linear model is reasonable for the situation described.

the price of a particular model car is \\$21,000 today and rises with time at a constant rate of \\$930 per year. how much will a new car of this model cost in 3.9 years?

select the correct choice below and fill in the answer box to complete your choice.
(simplify your answer.)

a. the independent variable is the price (p), in dollars, and the dependent variable is time (t), in years. the linear function that models this situation is \\(t = \\)

b. the independent variable is time (t), in years, and the dependent variable is the price (p), in dollars. the linear function that models this situation is \\(p = \\)

Explanation:

Identify the variables

Using the Independent and Dependent Variables knowledge point

The problem states that the price of a car rises with time. Time \(t\) (in years) is the independent variable, and the price \(p\) (in dollars) is the dependent variable. Thus, option B is correct.

Formulate the linear function

Using the Linear Modeling Conditions knowledge point

The initial price today is \(\$21,000\), which represents the vertical intercept. The price rises at a constant rate of \(\$930\) per year, which represents the slope.

$$ p = 21000 + 930t $$

Calculate the price in 3.9 years

Substitute \(t = 3.9\) into the linear model:

$$
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$$

Evaluate model reasonableness

A linear model assumes a constant rate of increase indefinitely. In reality, car prices fluctuate based on market demand, inflation, and model updates, meaning a linear model is only a reasonable approximation for a short period.

Answer:

• (A) The independent variable is the price (p), in dollars, and the dependent variable is time (t), in years. The linear function that models this situation is t = <blank></blank>
• (B) The independent variable is time (t), in years, and the dependent variable is the price (p), in dollars. The linear function that models this situation is p = <blank>\(21000 + 930t\)</blank> (Correct answer)