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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?

Respuesta :

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

[tex]m(t) =23,000^{(0.97)t}[/tex]

Therefore, in the first year we have

m(1) = $22310

Ī”m(tā‚) = $23000 -$22310 = $690

m(2 ) = $21640

Ī”m(t) = $23000 -$21640= $1359.3

Ī”m(tā‚‚ - tā‚) = $1359.3 Ā - $690 = $669.3

m(3) = $20991.479

Ī”m(tā‚ƒ) = $23000 -$20991.479= $2008.521

Ī”m(tā‚ƒ - tā‚‚) = Ā $2008.521 - $ $1359.3 = $649.221

m(4) = $20361.73

Ī”m(t) = $23000 -$20361.73= $2368.265

Ī”m(tā‚„ - tā‚ƒ) = $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;

[tex]\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)[/tex]

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