A car was valued at $38,000 in the year 1993. The value depreciated to $11,000 by the year 2004.
A) What was the annual rate of change between 1993 and 2004?
r
=
Round the rate of decrease to 4 decimal places.
B) What is the correct answer to part A written in percentage form?
r
=
%.
C) Assume that the car value continues to drop by the same percentage. What will the value be in the year 2007 ?
value = $
Round to the nearest 50 dollars
A) To find the annual rate of change between 1993 and 2004, we can use the formula for the rate of change:
Rate of change = (Final value - Initial value) / Number of years
In this case, the initial value is $38,000 in 1993 and the final value is $11,000 in 2004. The number of years between 1993 and 2004 is 2004 - 1993 = 11 years. Plugging these values into the formula:
Rate of change = ($11,000 - $38,000) / 11
Simplifying this equation, we get:
Rate of change = -$27,000 / 11
Rounding this answer to 4 decimal places, we have:
Rate of change = -2454.5455
So, the annual rate of change between 1993 and 2004 is approximately -2454.5455.
B) To convert the annual rate of change to percentage form, we simply multiply it by 100:
Rate of change in percentage = -2454.5455 * 100
Simplifying this equation, we get:
Rate of change in percentage ≈ -245,454.55%
So, the correct answer to part A written in percentage form is approximately -245,454.55%.
C) To find the value of the car in the year 2007, we can assume that the car value continues to drop by the same percentage as determined in part A.
In the year 2004, the value of the car was $11,000. To find the value in 2007, we need to determine how many years have passed since 2004.
2007 - 2004 = 3 years
Using the annual rate of change of approximately -2454.5455% from part A, we can calculate the value in 2007 as follows:
Value in 2007 = $11,000 * (1 + (-2454.5455 / 100)) ^ 3
Simplifying this equation, we get:
Value in 2007 = $11,000 * (0.0075454545) ^ 3
Calculating this value, we find:
Value in 2007 ≈ $9.22
Therefore, the value of the car in the year 2007 would be approximately $9.22. Rounded to the nearest $50, it would be $0.
2004 - 1993 = 11 years
11,000 = 38,000 *x^11
x^11 = 11/38 = 0.289473684
11 log x = -0.538390911
log x = -0.048944628
x = .893419385
r = 1-x = 0.106580614 = 0.1066 or down 10.66 % per year
2007 - 1993 = 14 years old
38,000 * .893419385^14 = 7844.28 = $7,850