https://file.aiquickdraw.com/mathbot/file/1727829164499raxzd6ns.jpg

Let's analyze the problem based on the image provided. Here's a breakdown of the situation:

### Given information:
- The company manufactures finished furniture ($ x $) and unfinished furniture ($ y $).
- The price-demand functions for finished ($ p $) and unfinished furniture ($ q $) are given as:
  \[
  p = 200 - \frac{1}{5}x - \frac{1}{10}y
  \]
  \[
  q = 160 - \frac{1}{10}x - \frac{1}{4}y
  \]
  where $ p $ is the price for finished units and $ q $ is the price for unfinished units.

- The total revenue function $ R(x, y) $ is the sum of revenue from finished and unfinished furniture, defined as:
  \[
  R(x, y) = p \cdot x + q \cdot y
  \]

### Required:
You need to compute the total revenue for:
1. $ R(100, 80) $
2. $ R(80, 100) $

#### Step 1: Calculate $ p $ and $ q $ for $ x = 100 $ and $ y = 80 $.
- $ p = 200 - \frac{1}{5}(100) - \frac{1}{10}(80) = 200 - 20 - 8 = 172 $
- $ q = 160 - \frac{1}{10}(100) - \frac{1}{4}(80) = 160 - 10 - 20 = 130 $

Now, calculate $ R(100, 80) $:
\[
R(100, 80) = 172 \cdot 100 + 130 \cdot 80 = 17200 + 10400 = 27600
\]

#### Step 2: Calculate $ p $ and $ q $ for $ x = 80 $ and $ y = 100 $.
- $ p = 200 - \frac{1}{5}(80) - \frac{1}{10}(100) = 200 - 16 - 10 = 174 $
- $ q = 160 - \frac{1}{10}(80) - \frac{1}{4}(100) = 160 - 8 - 25 = 127 $

Now, calculate $ R(80, 100) $:
\[
R(80, 100) = 174 \cdot 80 + 127 \cdot 100 = 13920 + 12700 = 26620
\]

### Interpretation:
- $ R(100, 80) = 27,600 $ indicates that the revenue from selling 100 finished pieces and 80 unfinished pieces is $27,600.
- $ R(80, 100) = 26,620 $ indicates that the revenue from selling 80 finished pieces and 100 unfinished pieces is $26,620.

Let me know if you'd like further clarification on any of the steps!

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Here are 5 related questions for deeper exploration:
1. How would changing the unit prices $ p $ and $ q $ affect the revenue function $ R(x, y) $?
2. What is the maximum possible revenue if only unfinished furniture is sold?
3. How does the interaction between $ x $ and $ y $ in the demand functions influence pricing strategy?
4. Could $ R(x, y) $ be optimized further, and what techniques could be used to find optimal sales quantities?
5. How would costs and profit be integrated into this model?

**Tip**: When calculating revenue functions, always carefully track how each variable (price and quantity) is related to avoid missteps in algebraic simplifications.

Country Workshop manufactures both finished and unfinished furniture for the home. The estimated quantities demanded each week are x and y units when the corresponding unit prices are p = 200 - (1/5)x - (1/10)y and q = 160 - (1/10)x - (1/4)y, respectively. For the weekly total revenue function R(x, y), compute the values of R(100, 80) and R(80, 100).

Learn how to calculate the total revenue function R(x, y) for both finished and unfinished furniture sales using the price-demand functions. This problem explores algebraic techniques and revenue modeling, suitable for high school students in Grades 11-12.

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