Select the correct answer. The manager at a car dealership is tracking the selling prices of two different used car models. When the tracking began, the selling price of model A was less than \( \$ 8,000 \), and the selling price of model \( B \) was at most \( \$ 10,000 \). The manager has determined that the price of model \( A \) is decreasing at a rate of \( 12 \% \) each year, and the price of model \( B \) is decreasing at a rate of \( 15 \% \) each year. Which system of inequalities can be used to determine after how many years, \( t \), that the selling price, \( y \), will be the same for both car models? A. \( \left\{\begin{array}{l}y \leq 8,000(0.88)^{t} \\ y<10,000(0.85)^{t}\end{array}\right. \) B. \( \left\{\begin{array}{l}y \leq 8,000(1.12)^{t} \\ y<10,000(1.15)^{t}\end{array}\right. \) C. \( \left\{\begin{array}{l}y<8,000(1.12)^{t} \\ y \leq 10,000(1.15)^{t}\end{array}\right. \) D. \( \left\{\begin{array}{l}y<8,000(0.88)^{t} \\ y \leq 10,000(0.85)^{t}\end{array}\right. \) Reset Next
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To determine the selling prices of both car models over time, we need to represent the decreasing prices as exponential functions. The price of model A decreases at a rate of 12% each year, thus its price can be represented as \( y = 8,000(0.88)^{t} \). For model B, with a 15% decrease, the price is represented as \( y = 10,000(0.85)^{t} \). With this understanding, the correct answer is option A: \( \left\{\begin{array}{l}y \leq 8,000(0.88)^{t} \\ y<10,000(0.85)^{t}\end{array}\right. \) This system of inequalities accurately captures the conditions for the prices of both models over time!