A baker will supply 25 jumbo cinnamon rolls to a cafe at a price of $4.40 each. If she is offered $4.10, then she will supply 2 fewer rolls to the cafe. The cafe's demand for jumbo cinnamon rolls is given by *p* = *D*(*x*) = -0.75*x* + 14.15. What is the equilibrium point?
 rolls at a price of $  each

To find the equilibrium point, we need to determine the price and quantity where the supply and demand are equal. Here's how to solve the problem step-by-step:

### Step 1: Determine the Supply Equation

From the problem statement:
- When the price $ p = 4.40 $, the quantity supplied $ x = 25 $.
- When the price $ p = 4.10 $, the quantity supplied $ x = 23 $.

We can use these two points to find the supply equation in the form $ p = S(x) = mx + b $.

1. Calculate the slope $ m $ of the supply function:
   \[
   m = \frac{p_2 - p_1}{x_2 - x_1} = \frac{4.10 - 4.40}{23 - 25} = \frac{-0.30}{-2} = 0.15
   \]

2. Use the slope and one of the points to find the y-intercept $ b $:
   \[
   p - 4.40 = 0.15(x - 25)
   \]
   Solving for $ b $:
   \[
   4.40 = 0.15 \times 25 + b
   \]
   \[
   4.40 = 3.75 + b
   \]
   \[
   b = 4.40 - 3.75 = 0.65
   \]

So, the supply equation is:
   \[
   p = 0.15x + 0.65
   \]

### Step 2: Equate the Supply and Demand Equations

The demand function is given as:
   \[
   p = D(x) = -0.75x + 14.15
   \]

Set the supply and demand equations equal to each other to find the equilibrium:
   \[
   0.15x + 0.65 = -0.75x + 14.15
   \]

### Step 3: Solve for $ x $

1. Combine like terms:
   \[
   0.15x + 0.75x = 14.15 - 0.65
   \]
   \[
   0.90x = 13.50
   \]

2. Divide by 0.90 to find $ x $:
   \[
   x = \frac{13.50}{0.90} = 15
   \]

### Step 4: Find the Equilibrium Price

Substitute $ x = 15 $ back into either the supply or demand equation to find the equilibrium price $ p $. Using the supply equation:
   \[
   p = 0.15 \times 15 + 0.65 = 2.25 + 0.65 = 2.90
   \]

### Equilibrium Point

The equilibrium point is where the quantity supplied equals the quantity demanded:
- **Quantity (x):** 15 rolls
- **Price (p):** $2.90

Would you like a more detailed explanation of any part of this process?

Here are 5 follow-up questions to deepen understanding:

1. What happens to the equilibrium price if the supply decreases?
2. How would the equilibrium change if the demand function had a steeper slope?
3. What is the interpretation of the slope in the supply equation?
4. How does this equilibrium point compare to other forms of equilibrium in economics?
5. What would happen to the equilibrium if a fixed tax were applied per roll?

**Tip:** Understanding how to derive the supply and demand equations is key to solving equilibrium problems in economics.

A baker will supply 25 jumbo cinnamon rolls to a cafe at a price of $4.40 each. If she is offered $4.10, then she will supply 2 fewer rolls to the cafe. The cafe's demand for jumbo cinnamon rolls is given by p = D(x) = -0.75x + 14.15. What is the equilibrium point?

Solve an equilibrium point problem involving a baker's supply of cinnamon rolls and a cafe's demand function. Learn how to derive the supply and demand equations, find the equilibrium price and quantity using algebraic methods. Suitable for high school students in Grades 10-12.

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