A car was valued at $35,000 in the year 1995. The value depreciated to $12,000 by the year 2006.
A)What was the annual rate of change between 1995 and 2006? (Round to 4 decimal places)
B) What is the correct answer to part A written in percentage form?
C)Assume that the car value continues to drop by the same percentage. What will the value be in the year 2011?
Please help ASAP and explain step by step the 3 questions!!! :(
A) To find the annual rate of change, we will use the formula for finding the rate of change between two points:
Rate of change = (final value - initial value) / number of years
In this case, the initial value is $35,000 in 1995 (t0) and the final value is $12,000 in 2006 (t1). The number of years between 1995 and 2006 is 2006 - 1995 = 11 years.
So, the rate of change = (12,000 - 35,000) / 11
= -23,000 / 11
= -2090.909
Therefore, the annual rate of change between 1995 and 2006 (rounded to 4 decimal places) is approximately -2090.909.
B) To express the rate of change in percentage form, we multiply the rate of change by 100:
Percentage rate of change = -2090.909 * 100
= -209,090.9%
Therefore, the correct answer to part A in percentage form is approximately -209,090.9%.
C) Since we assume that the car's value continues to drop by the same percentage, we need to determine the percentage decrease per year.
From part A, we found that the annual rate of change is approximately -2090.909. To find the percentage decrease, we divide it by the initial value and multiply by 100:
Percentage decrease per year = (-2090.909 / 35,000) * 100
= -5.975 \%
Therefore, the car's value is decreasing by approximately 5.975% per year.
To calculate the value of the car in the year 2011, we need to find the value after a total of 16 years (2011 - 1995 = 16 years).
Using the formula for compound interest, we can calculate the future value:
Value in 2011 = Initial value * (1 - percentage decrease per year)^number of years
= $35,000 * (1 - 0.05975)^16
= $35,000 * (0.94025)^16
= $35,000 * 0.568
= $19,880
Therefore, the value of the car in the year 2011 would be approximately $19,880.
(12/35)^(1/11) = 0.9072
see what you can do with that ...