An online furniture store sells chairs for $150 each and tables for $550 each. Every day, the store can ship no more than 32 pieces of furniture and must sell no less than $6800 worth of chairs and tables. If 30 chairs were sold, determine the minimum number of tables that the store must sell in order to meet the requirements.

To solve this problem, let's break it down step by step.

### Step 1: Define the variables and the problem
- Let $ x $ represent the number of chairs sold.
- Let $ y $ represent the number of tables sold.

Given:
- Each chair is sold for $150.
- Each table is sold for $550.
- The store can ship no more than 32 pieces of furniture each day, so:
  \[
  x + y \leq 32
  \]
- The store must sell at least $6800 worth of chairs and tables, so:
  \[
  150x + 550y \geq 6800
  \]
- It is given that 30 chairs were sold, so $ x = 30 $.

### Step 2: Substitute the value of $ x $ into the inequalities
Substitute $ x = 30 $ into the inequalities:

1. **Furniture constraint:**
   \[
   30 + y \leq 32
   \]
   Simplifying:
   \[
   y \leq 2
   \]

2. **Revenue constraint:**
   \[
   150(30) + 550y \geq 6800
   \]
   Simplifying the equation:
   \[
   4500 + 550y \geq 6800
   \]
   Subtract 4500 from both sides:
   \[
   550y \geq 2300
   \]
   Now, divide by 550:
   \[
   y \geq \frac{2300}{550} \approx 4.18
   \]

Since $ y $ must be an integer, round up to the nearest whole number:
\[
y \geq 5
\]

### Step 3: Conclusion
Given that $ y \leq 2 $ from the first inequality and $ y \geq 5 $ from the second inequality, **it's impossible to meet the conditions** if 30 chairs were sold. The second condition requires selling at least 5 tables to meet the revenue requirement, but this would exceed the 32-piece shipping limit when combined with 30 chairs.

Thus, under the given conditions, selling 30 chairs makes it impossible to satisfy both the shipping and revenue constraints simultaneously.

Would you like to explore this further, or do you have any other questions? Here are five related questions you might find interesting:

1. What happens if the store sells fewer than 30 chairs?
2. How does the result change if the store can ship more than 32 pieces?
3. What if the price of tables changes—how would that affect the minimum number required?
4. How would the solution differ if the revenue requirement were reduced?
5. What if the store introduces a new item—how would that affect the constraints?

**Tip:** When dealing with inequalities, always check the logical consistency of multiple constraints, as they may not always be simultaneously satisfiable.

Learn how to solve a linear inequalities problem in a furniture sales context where the store sells chairs and tables. This problem involves constraints on shipping and revenue, suitable for advanced high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation