Final answer:
To calculate the annual depreciation rate x% for a car, an equation using the formula for exponential decay is set up with the car's initial value and its value after three years. Solving this equation gives the annual depreciation rate.
Step-by-step explanation:
To find the annual depreciation rate x%, we must understand that the value of the car decreases by the same percentage each year over three years. The initial value of the car V₀ is £15000, and the value after three years V₃ is £9540.85. We can use the formula for exponential decay which is V₃ = V₀(1 - x/100)^3.
First, we express the problem as an equation: 9540.85 = 15000(1 - x/100)^3. By dividing both sides by the initial value, we get 9540.85 / 15000 = (1 - x/100)^3. Next, we calculate the left side and take the cube root of both sides to isolate (1 - x/100). We find that the cube root of 9540.85 / 15000 is approximately equal to 1 - x/100.
After calculating the above value, we subtract it from 1 and convert the result to a percentage to get the value of x. This yields the annual depreciation rate.