Gia has just purchased a car whose value in dollars, because of depreciation, will be worth C=25,000e−0.04t after t months have passed. At the same time, she is saving $100 per month. (Interest on her savings can be ignored.) Which of the following best represents when the combined value of her car and savings will start to increase with time?
The combined value of her car and savings will start to increase with time when the total value (car value + savings) is greater than the initial value (car value at t = 0).
Let's denote the car value C and the savings S after t months.
Car value: C = 25,000e^(-0.04t)
Savings: S = 100t
The total value is then V = C + S = 25,000e^(-0.04t) + 100t.
To find when the total value starts to increase, we need to find the value of t for which V > C at t = 0.
V > C at t = 0 can be represented as 25,000e^(-0.04 * 0) + 100 * 0 > 25,000e^0
This simplifies to 25,000 > 25,000.
Since 25,000 is greater than 25,000, the combined value of her car and savings will start to increase at t = 0, which is immediately after Gia has purchased the car.