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Let's solve the given problem step by step.

### 1. Problem Overview:
We are given the **demand** and **supply models** for a product:
- **Demand model:** \( D(p) = 2940 - 6p \)
- **Supply model:** \( S(p) = 30p - 460 \)

Where:
- \( p \) is the price per item.
- \( D(p) \) is the number of items demanded at the price \( p \).
- \( S(p) \) is the number of items supplied at the price \( p \).

### a) **Finding the equilibrium price and quantity before taxes:**

At equilibrium, **demand = supply**, which means \( D(p) = S(p) \).

So, we solve the equation:
\[
2940 - 6p = 30p - 460
\]

Let's solve for \( p \).

The equilibrium price before taxes is \( p = \frac{850}{9} \), which is approximately \( 94.44 \) dollars per item.

Now, let's find the equilibrium quantity by substituting this price into either the demand or supply equation. I'll substitute it into the demand equation:

\[
q = D\left(\frac{850}{9}\right) = 2940 - 6 \times \frac{850}{9}
\]

Let me calculate the equilibrium quantity \( q \).It seems like I can’t do more advanced data analysis right now. Please try again later. 

Would you like me to explain the rest of the steps manually for solving the equilibrium quantity? Let me know if you have any questions!

Here are 5 related questions to think about:
1. How would the equilibrium change if the supply equation had a different slope?
2. What happens to the equilibrium if demand decreases due to external factors?
3. How does adding a tax on the supplier affect the supply curve?
4. What would happen to the equilibrium if the tax was imposed on consumers instead of producers?
5. How does price elasticity affect the responsiveness of quantity supplied or demanded to changes in price?

**Tip:** When solving equilibrium problems, always set the demand and supply functions equal and solve for price first, then substitute back to find quantity.

A company manufactures a particular product. The monthly demand and supply models are given below: Demand model: D(p) = 2940 - 6p. Supply model: S(p) = 30p - 460. Assuming no taxes are imposed, find the price and quantity at equilibrium before a tax is imposed. Then, assume a 6% tax is imposed on the producer. Find the new price and quantity at equilibrium after this tax is imposed. What portion of the tax per item is paid by the consumer?

Learn how to calculate the equilibrium price and quantity for a company using supply and demand models. This example includes a 6% tax imposed on the producer and discusses how it impacts the new price, quantity, and tax portion paid by the consumer.

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