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? (1 point) Responses 28.8 months 28.8 months 40.2 months 40.2 months 57.6 months 57.6 months 74.9 months 74.9 months 115.1 months
To determine when the combined value of Gia's car and savings will start to increase with time, we need to find the point where the value of the car equals the cumulative amount she has saved.
The value of the car after t months is given by C = 25,000e^(-0.04t).
The cumulative amount Gia has saved after t months is given by 100t.
To find when the combined value starts to increase, we need to find the point where C = 100t.
Substituting the given expression for C, we have 25,000e^(-0.04t) = 100t.
Divide both sides of the equation by 100t to isolate the exponential term:
25,000e^(-0.04t) / 100t = 1.
Simplify the left side:
250e^(-0.04t) / t = 1.
Multiply both sides by t:
250e^(-0.04t) = t.
To solve this equation, we can use trial and error, or use numerical methods such as graphing or using a calculator.
Using a graphing calculator or a graphing software, we find that the solution to the equation is approximately t = 57.6 months.
Therefore, the combined value of Gia's car and savings will start to increase at approximately 57.6 months.
The correct response is:
57.6 months.