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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

Ask by Herrera Kelley. in the United States
Feb 03,2025

Real Tutor Solution

Tutor-Verified Answer

Answer

The correct system of inequalities is: \[ \left\{ \begin{array}{l} y < 8,000(0.88)^t \\ y \leq 10,000(0.85)^t \end{array} \right. \] So, the answer is **D**.

Solution

To determine when the selling prices of both car models \( A \) and \( B \) will be the same after \( t \) years, let's analyze the given information: 1. **Initial Prices:** - **Model A:** The initial price is **less than** \$8,000. - **Model B:** The initial price is **at most** \$10,000. 2. **Depreciation Rates:** - **Model A:** Decreases at a rate of **12%** each year, so the price after \( t \) years is \( 8000 \times (0.88)^t \). - **Model B:** Decreases at a rate of **15%** each year, so the price after \( t \) years is \( 10000 \times (0.85)^t \). 3. **Setting Up the Inequalities:** - Since Model A's initial price was **less than** \$8,000, after \( t \) years, its price will be **less than** \( 8000 \times (0.88)^t \). Therefore, we have: \[ y < 8000 \times (0.88)^t \] - Since Model B's initial price was **at most** \$10,000, after \( t \) years, its price will be **at most** \( 10000 \times (0.85)^t \). Therefore, we have: \[ y \leq 10000 \times (0.85)^t \] 4. **System of Inequalities:** Combining both inequalities, the system that determines when the selling prices will be the same is: \[ \left\{\begin{array}{l} y < 8000(0.88)^t \\ y \leq 10000(0.85)^t \end{array}\right. \] **Correct Answer:** **D.** \( \left\{\begin{array}{l}y<8,000(0.88)^{t} \\ y \leq 10,000(0.85)^{t}\end{array}\right. \)

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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!

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