Demand for widgets is a linear function of price.  If the price of Widgets is set to $20, 4200 widgets will be sold.  If the price is raised to $25, demand falls to 3600 widgets. ) Find the demand function D(x), where x represents the price and D(x) =  y represents the demand for widgets. Find the revenue function R(x), which is price times the demand

To find the demand function $ D(x) $ and the revenue function $ R(x) $ for the widgets, we'll follow these steps:

### 1. Finding the Demand Function $ D(x) $

Given:
- When the price $ x = 20 $, demand $ D(x) = 4200 $.
- When the price $ x = 25 $, demand $ D(x) = 3600 $.

The demand function is linear, so we can write it in the form:
\[ D(x) = mx + b \]
where $ m $ is the slope and $ b $ is the y-intercept.

#### Finding the Slope $ m $

The slope $ m $ of a linear function is given by:
\[ m = \frac{\Delta y}{\Delta x} = \frac{D(25) - D(20)}{25 - 20} \]

Plugging in the values:
\[ m = \frac{3600 - 4200}{25 - 20} = \frac{-600}{5} = -120 \]

#### Finding the Y-Intercept $ b $

Now that we have the slope $ m = -120 $, we can use one of the points to find the y-intercept $ b $. Let's use the point $(20, 4200)$:

\[ 4200 = -120(20) + b \]
\[ 4200 = -2400 + b \]
\[ b = 4200 + 2400 = 6600 \]

So, the demand function is:
\[ D(x) = -120x + 6600 \]

### 2. Finding the Revenue Function $ R(x) $

Revenue $ R(x) $ is given by the product of price $ x $ and demand $ D(x) $:
\[ R(x) = x \cdot D(x) \]

Substitute the demand function $ D(x) = -120x + 6600 $ into the revenue function:
\[ R(x) = x(-120x + 6600) \]
\[ R(x) = -120x^2 + 6600x \]

### Summary

- **Demand function:** $ D(x) = -120x + 6600 $
- **Revenue function:** $ R(x) = -120x^2 + 6600x $

Would you like more details or have any questions?

Here are 5 questions you might want to ask:

1. How do you interpret the slope in the demand function?
2. What does the y-intercept represent in the context of the demand function?
3. How can you find the price that maximizes revenue?
4. What is the significance of the revenue function's quadratic term?
5. How would you determine the break-even points for this revenue function?

**Tip:** When dealing with linear functions, always identify two key points and use them to determine the slope and y-intercept.

Learn how to find the demand function and revenue function for widgets based on given price-demand data. This problem covers linear functions, slope-intercept form, and revenue calculation. Suitable for students in Grades 10-12.

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