Question

The table below shows the value of a car ($ in thousands) and the time (in years) since it was purchased. Find the average rate of change in value from years 2 to 4.
Time (years) | Car's value (thousands)
0 | 32
1 | 27.2
2 | 23.1
3 | 19.65
4 | 16.70
A. -$3,450 per year
B. -$3,200 per year
C. -$6400 per year
D. -$3.20 per year

Answer 1

To find the average rate of change in the value of the car from year 2 to year 4, we'll use the formula for the average rate of change, which is:

\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
\]

where \( f(a) \) is the value of the function at time \( a \) (the starting point), and \( f(b) \) is the value of the function at time \( b \) (the ending point). In this case:

- \( a = 2 \)
- \( b = 4 \)
- \( f(2) = 23.1 \) (the value at year 2)
- \( f(4) = 16.70 \) (the value at year 4)

Now, substitute the values into the formula:

\[
\text{Average Rate of Change} = \frac{16.70 - 23.1}{4 - 2}
\]

Calculating the numerator:

\[
16.70 - 23.1 = -6.4
\]

Calculating the denominator:

\[
4 - 2 = 2
\]

Now, substitute these into the equation:

\[
\text{Average Rate of Change} = \frac{-6.4}{2} = -3.2
\]

Thus, the average rate of change in value from year 2 to year 4 is -3.2 thousand dollars per year.

Therefore, the answer is **D. -$3.20 per year**.