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 $\$ 12,000$ today and rises with time at a constant rate of $\$ 820$ 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 time ( $t$ ), in years, and the dependent variable is the price (p), in dollars. The linear function that models this situation is $p=\square$.
B. The independent variable is the price (p), in dollars, and the dependent variable is time $(\mathrm{t})$, in years. The linear function that models this situation is $\mathrm{t}=\square$.
Answer
Final Answer: The price of the car in 3.9 years will be \(\boxed{15198}\).
Step 1 ：The problem is asking for the price of a car model in 3.9 years given that the price today is $12,000 and it increases at a constant rate of $820 per year. This is a linear function problem where the independent variable is time (t) in years and the dependent variable is the price (p) in dollars.
Step 2 ：The linear function can be represented as \(p = mt + b\) where m is the slope (rate of increase per year) and b is the y-intercept (initial price). In this case, \(m = 820\) and \(b = 12000\).
Step 3 ：We need to find the price in 3.9 years, so we substitute \(t = 3.9\) into the equation and solve for p.
Step 4 ：Substituting the values into the equation, we get \(p = 820 * 3.9 + 12000\).
Step 5 ：Solving the equation, we get \(p = 15198\).
Step 6 ：Final Answer: The price of the car in 3.9 years will be \(\boxed{15198}\).
