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?
−660
660
2638
−2638
its 660
To find the average rate of change of the car's value in dollars per year during the first 4 years Kenzie owns it, we need to find the change in value over 4 years and divide it by 4.
First, we find the initial value of the car by plugging t = 0 into the function:
m(0) = 23,000(0.97)^0
m(0) = 23,000(1)
m(0) = 23,000
Next, we find the value of the car after 4 years by plugging t = 4 into the function:
m(4) = 23,000(0.97)^4
m(4) = 23,000(0.8731814)
m(4) ≈ 20,080.14
The change in value over 4 years is the difference between the final value and the initial value:
Change in value = 20,080.14 - 23,000
Change in value ≈ -2,919.86
Finally, we calculate the average rate of change by dividing the change in value by 4:
Average rate of change = -2,919.86 / 4
Average rate of change ≈ -729.97
Rounded to the nearest dollar, the average rate of change of the car's value in dollars per year during the first 4 years Kenzie owns it is approximately -$730. Therefore, the answer is −660.
To find the average rate of change of the car's value, we need to find the difference in the car's value at the end and the beginning of the given time period, and then divide that difference by the length of the time period.
In this case, the time period is the first 4 years. So, we need to calculate the difference in the car's value after 4 years and after 0 years.
To find the value after 4 years, we substitute t = 4 into the function m(t) = 23,000(0.97)t:
m(4) = 23,000(0.97)^4
m(4) ≈ 23,000(0.8836) ≈ 20,328.8
To find the value after 0 years, we substitute t = 0 into the function:
m(0) = 23,000(0.97)^0
m(0) = 23,000(1) = 23,000
Now, we can find the difference in the car's value:
Difference = m(4) - m(0)
Difference = 20,328.8 - 23,000
Difference ≈ -2,671.2
Finally, we divide the difference by the length of the time period, which is 4 years:
Average Rate of Change = Difference / Time period
Average Rate of Change = -2,671.2 / 4 ≈ -667.8
Rounding this to the nearest dollar, we get an average rate of change of -668 dollars per year.
Therefore, the answer is −660.