If a new car is valued at $21000 and 10 years later it is valued at $3000, then what is the average rate of change of its value during those 10 years?

Question

I can't seem to get the answer. Will You please help me answer this question. I am studying for a test.

Answer 1

"If a new car is valued at $21000 and 10 years later it is valued at $3000, then what is the average rate of change of its value during those 10 years?"

 

The Average Rate of Change of a function f(x) between x = x1 and x = x2 is defined as:

 

Δy/Δx = (f(x2) - f(x1))/(x2 - x1). [See any precalculus or calculus I text.]

 

For this question:

 

Δy/Δx = (3000 - 21000)/(10 - 0) = –18000/10 = –$1800/year.

 

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Δy/Δx is the slope of the Secant Line between P1 and P2. If P2 is moved along the function graph closer and closer to P1, the secant line they define will approach the tangent line at P1. The slope of the tangent line is the slope of the function at P1 and the derivative of f at P1.

Answer 2

Most likely, the car loses a percentage of its value every year.

If that is the case, then we should use the formula for exponential decay.

 

X = X_o(1-A)^t

 

X_o is the original value

A is the percentage lost every year

t is the number of years

X is the new value

 

3000 = 21000(1-A)¹⁰

1/7 = (1-A)¹⁰

1/7^(1/10) = (1-A) 

1-A = .823

A  = 17.7% value lost per year

 

In order to visualize what is happening...

Initial value : 21000

After 1 year: 21000*.823 = 17283

After 2 years: 17283*.823 = 14223

...

After 10 years: 21000*.823¹⁰

 

 

Hmm, although I do think that the way the question is worded, this question is much simpler than I've made it out to be, and Eugene's answer is correct.

Answer 3

21,000 - 3,000 = 18,000

 

Divide by the number of years:

 

18,000 / 10 = Average rate of change: $1,800

Answer 4

Hi Melissa,

             There are several ways to answer this depending on how you interpret average rate of change. I'll answer it first using the simplest .

 

The  change in value =3000 -21000 =-18000 in 10 years so the average rate of change is -18000/10 =-1800 $/year. Stated in words the car has lost $1,800 in value per year.

 

You can also say that the percent change in value is 100*(3000-21000)/21000 =85.71% so the average percent decline in value is 8.57%/year.

 

Hope this helps, let me know if this is what you were looking for.

 

Jim