3. A car cost $32 600 in the year 2001. By 2007, it is worth $12 295. By what percent did the car depreciate (decline in value) per year?
32600(1-R)^6=12295
correct?
r=?
WRONG
To solve this problem, we need to find the rate of depreciation per year. We can use the formula for compound interest or depreciation to find the value of the car in 2007:
A = P(1 - r)^n
Where:
A = Future value (12295)
P = Present value (32600)
r = Rate of depreciation per year (unknown)
n = Number of years (6)
Plugging in the given values and the formula, we have:
12295 = 32600(1 - r)^6
Now we can solve for r. First, divide both sides of the equation by 32600:
(1 - r)^6 = 12295 / 32600
Simplifying the right side of the equation:
(1 - r)^6 = 0.377668
Next, we need to take the 6th root of both sides to isolate (1 - r):
(1 - r) = (0.377668)^(1/6)
Now, solve for r by subtracting 1 from both sides:
- r = (0.377668)^(1/6) - 1
Finally, calculate r:
r ≈ -0.207899
So, the car has depreciated by approximately 20.79% per year.