The table below shows the value in dollars of a car at the end of `x` years. Number of Years, x: 0, 1, 2, 3. Value, v(x) (dollars): 11,000, 9,350, 7,948, 6,755. Which exponential function models this situation?
We can observe that the value of the car decreases over time. Additionally, the values in the table appear to be decreasing at a constant rate. This suggests that the relationship between the number of years and the value of the car is exponential.
To find the exponential function that models this situation, we can write it in the form:
v(x) = a * b^x,
where v(x) is the value of the car after x years, and a and b are constants.
From the table, we can see that when x = 0, v(x) = 11,000. This gives us the equation:
11,000 = a * b^0,
11,000 = a * 1,
a = 11,000.
Now we have:
v(x) = 11,000 * b^x.
To find the value of b, we can use the other values in the table. When x = 1, v(x) = 9,350. This gives us the equation:
9,350 = 11,000 * b^1,
9,350/11,000 = b^1,
0.85 = b.
Therefore, the exponential function that models this situation is:
v(x) = 11,000 * 0.85^x.