Answer:
The function that represents the value of the car after t years, where the quarterly rate of change can be found from a constant in the function is:
V(t) = $22,000 * (1 - 0.15/4)^4t
The quarterly rate of change is 3.51%.
Explanation:
Given:
The value of a brand new car is $22,000 and the value depreciates 15% every year.
To find:
Write a function to represent the value of the car after t years, where the quarterly rate of change can be found from a constant in the function.
Determine the percentage rate of change per quarter, to the nearest hundredth of a percent.
Solution:
The formula to calculate the value of the car after t years can be written as:
V(t) = $22,000 * (1 - 0.15/4)^(4t)
where V(t) is the value of the car after t years, and (1 - 0.15/4) is the quarterly rate of change. We raise this to the power of 4t since there are four quarters in a year, and we need to compound the quarterly depreciation rate t times.
To determine the percentage rate of change per quarter, we need to first find the quarterly depreciation rate. We can do this by dividing the annual depreciation rate by the number of quarters in a year:
Quarterly depreciation rate = 0.15 / 4 = 0.0375
To find the percentage rate of change per quarter, we multiply the quarterly depreciation rate by 100 and round to the nearest hundredth of a percent:
Percentage rate of change per quarter = 0.0375 * 100 = 3.75% (rounded to the nearest hundredth)
Therefore, the percentage rate of change per quarter is 3.75%.
Answer:
V(t) = $22,000 * (1 - 0.15/4)^(4t) rounded to four decimal places.
Percentage rate of change per quarter = 3.75%.