Question

A car was valued at $44,000 in the year 1993. The value depreciates to $15,000 by year 2002.

A) what was annual rate of change between 1993 and 2002?

B) Assume the car value continues to drop by same percentage. What will value be in year 2007?

Answer 1

A) -$29000/9 years

B) The car has lost its value by year 2007

Explanation:

The annual rate of change is given by:
`\\=(15000-44000)/(2002-1993)  \\=(-29,000)/(9) \\=-$3222.22`

Therefore, the equation of the line that describes this change of value in terms of years passed, is a line with the slope given by this rate of change, and that passes through the value $44000 at year zero. That is:

`y(x)=-3222.22x+44000`

In the year 2007, 2007-1993= 14 years have gone by, so the car's value can be obtained by replacing x with "14" in the formula for the line:

`y(14)=-3222.22(14)+44000\\y=-1111.11`

which being a negative number, means that the car has lost its value.

Answer 2

The car will have lost it's total value by 2007.

Explanation:

If initially the car was valued at 44,000$, and after 9 years it's value dropped to 15,000$, we can say that the car's value dropped in 29,000$. If we suppose that the drop is the same every year, we can say that it was of 3,222,2$ by each year.

This amount of money is the 7,3% of the initial value of the car (I multiplied 3,222,2 x 100 : 44,000).

a) The annual rate of change was of 7,3%.

b) There are 14 years between 1993 and 2007. If we multiply 7,3% by 14, we get that the car lost 102,2% of it's initial value.