Question

let v be the value (in thousands of dollars) of a car when it is t years old. some pairs of values of t and v are listed in the following table.
t years | 1 | 3 | 5 | 7 | 9
v thousands of dollars | 18 | 14 | 10 | 6 | 2
complete parts a to e.
b. estimate the age of the car when it is worth $12 thousand.
the age of the car is \\(\square\\) years when it is worth $12 thousand.

Explanation:

Step1: Analyze the data pattern

From the table, when \( t = 1 \), \( v = 18 \); \( t = 3 \), \( v = 14 \); \( t = 5 \), \( v = 10 \); \( t = 7 \), \( v = 6 \); \( t = 9 \), \( v = 2 \). The change in \( t \) is \( 2 \) (from 1 to 3, 3 to 5, etc.), and the change in \( v \) is \( - 4 \) (18 - 14 = 4, 14 - 10 = 4, etc.). So it's a linear relationship with slope \( m=\frac{\Delta v}{\Delta t}=\frac{-4}{2}=-2 \). The equation can be approximated as \( v - 18=-2(t - 1) \), or \( v=-2t + 20 \).

Step2: Solve for \( t \) when \( v = 12 \)

Substitute \( v = 12 \) into the linear equation \( 12=-2t + 20 \).
Subtract 20 from both sides: \( 12-20=-2t \), so \( - 8=-2t \).
Divide both sides by - 2: \( t = 4 \). We can also check the pattern: between \( t = 3 \) (v=14) and \( t = 5 \) (v=10), the value 12 is halfway between 14 and 10, so \( t \) should be halfway between 3 and 5, which is 4.

Answer:

4