Question 1172984: A used furniture store buys old mattresses for $50 and sells them for $200. Write an inequality representing all the possible combinations of mattress purchases and sales that will result in a profit of less than $1500. Graph the inequality, and describe an aspect of the used furniture business that an inequality cannot account for.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Let's break down this problem step-by-step.
**1. Define Variables**
* Let 'x' be the number of mattresses purchased.
* Let 'y' be the number of mattresses sold.
**2. Calculate Profit per Mattress**
* Selling price: $200
* Purchase price: $50
* Profit per mattress: $200 - $50 = $150
**3. Write the Profit Inequality**
* Total profit from sales: 150y
* Total cost of purchases: 50x
* Net profit: 150y - 50x
* We want the net profit to be less than $1500:
* 150y - 50x < 1500
**4. Simplify the Inequality**
* Divide the inequality by 50:
* 3y - x < 30
**5. Rewrite in Slope-Intercept Form (for graphing)**
* Add x to both sides:
* 3y < x + 30
* Divide by 3:
* y < (1/3)x + 10
**6. Graph the Inequality**
* **Graph the boundary line:** y = (1/3)x + 10
* Y-intercept: (0, 10)
* Slope: 1/3 (rise 1, run 3)
* **Dashed Line:** Since the inequality is strictly "less than" (<), use a dashed line to indicate that points on the line are *not* included in the solution.
* **Shade the Region:** Shade the region *below* the dashed line, as this represents y < (1/3)x + 10.
* **First Quadrant:** Remember that x and y must be non-negative (you can't buy or sell a negative number of mattresses), so the solution is restricted to the first quadrant.
**7. Aspect the Inequality Cannot Account For**
The inequality only accounts for the financial relationship between purchases, sales, and profit. It *cannot* account for:
* **Demand:** The inequality doesn't consider if there are enough customers willing to buy the mattresses. You might buy many mattresses but not be able to sell them all.
* **Condition of Mattresses:** The inequality assumes all mattresses are sold for $200. Some mattresses might be in poor condition and need repairs, reducing profit, or they might not sell at all.
* **Storage Costs:** The inequality does not factor in the cost of storing the mattresses.
* **Time:** The inequality doesn't consider the time it takes to buy and sell the mattresses. If mattresses sit in storage for a long time, the opportunity cost increases.
* **Market Fluctuations:** The inequality assumes a fixed selling price of $200. Market prices for used mattresses can change.
* **Labor Costs:** The inequality does not factor in the labor costs associated with moving, cleaning, or repairing the mattresses.
* **Disposal Costs:** Some mattresses might be too damaged to sell, and disposing of them could incur a cost.
* **Inventory Limits:** The store may only have so much space for mattresses.
|
|
|
