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

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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. $$ \left\{\begin{array}{l}y \leq 8,000(0.88)^{t} \ y<10,000(0.85)^{t}\end{array}ight. $$ B. $$ \left\{\begin{array}{l}y \leq 8,000(1.12)^{t} \ y<10,000(1.15)^{t}\end{array}ight. $$ C. $$ \left\{\begin{array}{l}y<8,000(1.12)^{t} \ y \leq 10,000(1.15)^{t}\end{array}ight. $$ D. $$ \left\{\begin{array}{l}y<8,000(0.88)^{t} \ y \leq 10,000(0.85)^{t}\end{array}ight. $$ Reset Next

UpStudy AI · Accepted Answer

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.** $$ \left\{\begin{array}{l}y<8,000(0.88)^{t} \ y \leq 10,000(0.85)^{t}\end{array}\right. $$
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**.

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