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Let's break down the information given and solve this step-by-step:

1. **Minimum and Maximum Order**:
   - Minimum chairs: 270
   - Maximum chairs: 440

2. **Pricing**:
   - If 270 chairs are bought, the price is $87 per chair.
   - For every chair above 270, there’s a discount of $0.25 per chair on the entire order.

3. **Objective**:
   - (a) Find the **largest revenue** the supplier can make.
   - (b) Find the **least revenue** the supplier can make.

### Solution Approach

Let $ x $ be the number of chairs ordered above the minimum 270. Therefore, the total chairs ordered is $ 270 + x $, where $ 0 \leq x \leq 170 $ (since the maximum number of chairs is 440).

The price per chair, given the discount, becomes:
\[
\text{Price per chair} = 87 - 0.25x
\]

The **revenue**, $ R $, is then calculated by:
\[
R = \text{Total Chairs} \times \text{Price per Chair} = (270 + x)(87 - 0.25x)
\]

#### Part (a): Maximum Revenue
To maximize the revenue, substitute $ x = 0 $ (for no additional chairs) and $ x = 170 $ (for the maximum chairs).

1. **When $ x = 0 $:**
   \[
   R = 270 \times 87 = 23490
   \]

2. **When $ x = 170 $:**
   \[
   \text{Price per chair} = 87 - 0.25 \times 170 = 44.5
   \]
   \[
   R = 440 \times 44.5 = 19580
   \]

So, the **maximum revenue** is **$23,490**.

#### Part (b): Minimum Revenue
The minimum revenue happens at the maximum discount applied, which is when $ x = 170 $, resulting in a revenue of **$19,580**.

### Answers
- (a) Largest revenue: **$23,490**
- (b) Least revenue: **$19,580**

Would you like further details or clarification on this solution?

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Here are some related questions to consider:
1. How would the revenue change if the discount was $0.50 per additional chair instead?
2. What if the minimum order quantity was raised to 300 chairs?
3. How does the revenue formula change if a fixed discount is applied regardless of the number of chairs?
4. Could the revenue model be maximized with non-integer values of $ x $?
5. What is the impact on revenue if the initial price per chair was different?

**Tip:** Always double-check units and ensure each step aligns with the conditions in the problem for accurate results.

An office building owner agrees to buy a minimum of 270 chairs and up to 440 chairs from a supplier. The price will be $87 per chair if only 270 chairs are bought, but will be discounted by $0.25 per chair (on the entire order) for every chair ordered in addition to the minimum. What is the largest revenue and least revenue the supplier can make under this deal?

Solve this revenue calculation problem with a discount applied to bulk chair orders. Learn how to find the largest and least revenue for a chair supplier with a minimum order of 270 chairs and a maximum of 440, using algebraic methods and discounting strategies. Suitable for high school students in Grades 10-12.

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