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

complete parts a to e.

the age of the car is 4 years when it is worth $12 thousand.

c. estimate the value of the car when it is 2 years old.
the value of the car is $16 thousand when it is 2 years old.

d. what is the v-intercept of the model? what does it mean in this situation?
select the correct choice and fill in the answer box to complete your choice.
(type an ordered pair.)
a. the v-intercept is. the v-intercept estimates the value of the new car.
b. the v-intercept is. the v-intercept estimates the age of the car when its value will be 0.

Explanation:

Step1: Identify the relationship type

The data shows a linear relationship (constant decrease: from t=1 to t=3, v decreases by 4; t=3 to t=5, also 4, etc.). So we can model it with a linear equation \( v = mt + b \), where \( b \) is the v - intercept (when \( t = 0 \)).

Step2: Calculate the slope (m)

Using two points, e.g., (1, 18) and (3, 14). The slope \( m=\frac{v_2 - v_1}{t_2 - t_1}=\frac{14 - 18}{3 - 1}=\frac{-4}{2}=-2 \).

Step3: Find the equation of the line

Using point - slope form with (1, 18): \( v - 18=-2(t - 1) \). Simplify: \( v=-2t + 2 + 18=-2t + 20 \).

Step4: Find the v - intercept

The v - intercept occurs when \( t = 0 \). Substitute \( t = 0 \) into \( v=-2t + 20 \), we get \( v = 20 \). So the ordered pair is (0, 20). The v - intercept estimates the value of the new car (when t = 0, the car is new), so option A is correct.

Answer:

A. The v - intercept is \((0, 20)\). The v - intercept estimates the value of the new car.