The main objective of this question is to find the **revenue model** of the given equation as just a function with respect to **x**.

This question uses the concept of **revenue model**. A revenue model is a **blueprint** that outlines how a **startup** company will **generate** revenue or annual profit out of its **basic business operations.** **R****evenue** is a** blueprint** that outlines how a startup business would then **generate revenue** or annual profit out of its **standard daily operations**, as well as how it will cover **operating costs** and **expenses**.

## Expert Answer

We have to find the revenue model for the given expression. A** revenue model** is a **blueprint** that outlines how a **startup company** will generate revenue or annual profit out of its **basic business** operations. The **given expression** is:

\[p \space = \space – \space \frac{1}{6}x \space + \space 100 \]

We **know** that:

\[R \space = \space xp \]

**So**:

\[R \space = \space x (- \space \frac{1}{6}x \space + \space 100 ) \]

**Multiplying** $ x $ results in:

\[R \space = \space – \space \frac{1}{6}x^2 \space + \space 100 x \]

**Hence,** the **final answer** is:

\[R \space = \space – \space \frac{1}{6}x^2 \space + \space 100 x \]

## Numerical Answer

The **revenue model** for the given expression $ p = – \frac{1}{6}x + 100 $ where p is the price in dollars and the quantity of product sold is $ x $ :

\[R \space = \space – \space \frac{1}{6}x^2 \space + \space 100 x \]

## Example

Find the revenue model for the two expressions $ p = – \frac{1}{8}x + 120 $ and $ p = – \frac{1}{8}x ^2 + 220 $ \space where $ p $ is the price in dollars and the quantity of product sold is $ x $ .

We have to **find the revenue model** for the given expression which is:

\[p \space = \space – \space \frac{1}{8}x \space + \space 120 \]

**where** $ p $ is the price in **dollars** and the **quantity** of **product** **sold** is $ x $.

We **know** that:

\[R \space = \space xp \]

**So**:

\[R \space = \space x (- \space \frac{1}{8}x \space + \space 120 ) \]

**Multiplying** $ x $ results in:

\[R \space = \space – \space \frac{1}{8}x^2 \space + \space 120 x \]

**Hence,** the **final answer** is:

\[R \space = \space – \space \frac{1}{8}x^2 \space + \space 120 x \]

**Now** for the **second expression** which is:

\[p \space = \space – \space \frac{1}{8}x ^2 + 220 \]

**where** $ p $ is the **price in dollars** and the **quantity of product** sold is $ x $

We have to** find the revenue model** for the **given expression,** which is:

\[p \space = \space – \space \frac{1}{8}x^2 \space + \space 220 \]

We **know** that:

\[R \space = \space xp \]

**So**:

\[R \space = \space x (- \space \frac{1}{8}x^2 \space + \space 220 ) \]

**Multiplying** $ x $ results in:

\[R \space = \space – \space \frac{1}{8}x^3 \space + \space 220 x \]

Thus, the **final answer** is:

\[R \space = \space – \space \frac{1}{8}x^3 \space + \space 220 x \]