A question about the average change of value

Question

A machine that originally cost $15 600 has a value of $7500 at the end of 3 years. The same machine has a value of $280l at the end of 8 years. Find the average rate of change in value(depreciation) of the machine between it’s purchase at the end of three years. Interpret the sign of your answer. Need it fast!

Answer 1

A = Pe^rt where t = years, r= annual interest rate,

the formula for continous compounding growth or decay. If r<0 it means decay or depreciation

7500=16500e^3r

75/165 = e^3r

take natural logs of both sides

3r = ln(15/33)

divide by 3

r = ln(15/33)/3= -26.28% annual decay rate = average rate of annual rate of change over 1st 3 years since purchase.

2801 = 16500e^8r

r = ln(2801/16500)/8 = -16.98% annual decay rate

meaning of the negative sign is that the rate of changes shows decay not positive growth,

but they're different rates, although not that far apare. still, the exponential model looks wrong.

the other major choice is linear decay

find the slope of the line through the points (0, 16500) and (3, 7500) and (8, 2801)

slope = (16500-7500)/(0-3) = -3000

every year, for the 1st 3 years, there's a negative change of 3000

-3000/16500 = -30/165 = -6/33 = about 18.18%

or

= (7500-2801)/(3-8) = 4699/-5 = about -940 which doesn't = -3000

or (16500-2801)/-8 = about -1712

the decay rate is not linear

but the answer you want is probably about 18 or 26% decay rate

although these differences may just indicate the measurements or data points include significant errors, such as rounding the years to integers

or a quadratic equation (a downward opening parabola) would always work for any 3 points, then find the slope of the secant line for the points with t=0 and t=3. but that's the same as the rate of change for a linear decay rate of about 18%

this all assumes the $2801 is correct, as that last digit looks more like an "l" than a"1"

like a letter l, just before m, rather than the number one. could be typo? maybe just a comma or a period, as l and the punctuation marks are close on most keyboards.