Question

The decrease in the worth of Kenzie's car can be represented by the function m(t)=23,000(0.97)t, where t is the time in years, and m is the car's value in dollars. What is the average rate of change of the car's value in dollars per year (rounded to the nearest dollar) during the first 4 years Kenzie owns it?

Answer 1

Answer:

The average rate of change of the cars value during the first four years Kenzie owns it is $659.566 per year

Step-by-step explanation:

Here we have

$m(t) =23,000^{(0.97)t}$

Therefore, in the first year we have

m(1) = $22310

Δm(t₁) = $23000 -$22310 = $690

m(2 ) = $21640

Δm(t) = $23000 -$21640= $1359.3

Δm(t₂ - t₁) = $1359.3  - $690 = $669.3

m(3) = $20991.479

Δm(t₃) = $23000 -$20991.479= $2008.521

Δm(t₃ - t₂) =  $2008.521 - $ $1359.3 = $649.221

m(4) = $20361.73

Δm(t) = $23000 -$20361.73= $2368.265

Δm(t₄ - t₃) = $2368.265 - $ $2008.521= $629.744

Therefore the average rate of change of the cars value during the first four years Kenzie owns it is ($690 + $669.3 + $649.221 + $629.744)/4 = $659.566 per year.

Also by differentiating the function, we have;

$\frac{\mathrm{d} m(t) }{\mathrm{d} t} = \frac{\mathrm{d} \left (23,000^{(0.97)t} \right )}{\mathrm{d} x} = 23,000\cdot e^{t\cdot ln(0.97)}\cdot ln(0.97)$