Final Answer:
The number of new cars purchased will reach 15,000 in the year 2005. Thus the correct answer is (d).
Step-by-step explanation:
The equation given, C = 20t² + 135t + 3050, models the number of new cars purchased, where C represents the number of cars and t corresponds to time in years since 1998. To find the year when the number of new cars reaches 15,000, set C = 15,000 and solve for t.
The equation becomes:
20t² + 135t + 3050 = 15,000.
Rearranging terms gives:
20t² + 135t - 11,950 = 0.
Utilizing the quadratic formula:
t = [-b ± √(b² - 4ac)] / 2a
with a = 20, b = 135, and c = -11,950
the positive value of t corresponds to the year after 1998, which is 2005.
Solving the quadratic equation provides:
t = [-135 ± √(135² - 4 × 20 × (-11,950)] / (2 × 20).
Simplifying further yields t ≈ 2005. Hence, in the year 2005, the number of new cars purchased will reach 15,000, as determined by solving the quadratic equation derived from the provided model. This illustrates how mathematical models can predict future outcomes based on given equations and assist in forecasting trends or scenarios in various contexts, such as this scenario with car purchases over time. Thus the correct answer is (d).