Executives of a bread company decide to add a new line of whole wheat bread. Research indicates that consumers will buy about 600 loaves at $1.95 a loaf and 1,100 at $1.50 a loaf. The company can supply 460 loaves at $1.65 per loaf and 1,260 loaves at $1.94 per loaf. Furthermore, the baking staff reports that it costs $280 for 850 loaves and $555 for 2,200 loaves. Let the input be p, price per loaf, and the output be q, number of loaves.

Find the market equilibrium price and quantity. (Round your answers to two decimal places.)

To find the market equilibrium price and quantity, we need to derive the demand and supply equations from the given data and then find the intersection of these two equations.

### Step 1: Determine the Demand Equation

The demand equation relates the price $ p $ to the quantity demanded $ q_d $. We are given two points on the demand curve:
- At $ p = 1.95 $, $ q_d = 600 $
- At $ p = 1.50 $, $ q_d = 1100 $

We can use the slope-intercept form of a linear equation, $ p = m \cdot q_d + b $, to find the demand equation.

First, calculate the slope $ m_d $:

\[
m_d = \frac{1.95 - 1.50}{600 - 1100} = \frac{0.45}{-500} = -0.0009
\]

Now, use the slope and one point to solve for the intercept $ b_d $. Using the point $ (q_d, p) = (600, 1.95) $:

\[
1.95 = -0.0009 \cdot 600 + b_d
\]

\[
b_d = 1.95 + 0.54 = 2.49
\]

So the demand equation is:

\[
p = -0.0009q_d + 2.49
\]

### Step 2: Determine the Supply Equation

The supply equation relates the price $ p $ to the quantity supplied $ q_s $. We are given two points on the supply curve:
- At $ p = 1.65 $, $ q_s = 460 $
- At $ p = 1.94 $, $ q_s = 1260 $

Again, calculate the slope $ m_s $:

\[
m_s = \frac{1.94 - 1.65}{1260 - 460} = \frac{0.29}{800} = 0.0003625
\]

Now, use the slope and one point to solve for the intercept $ b_s $. Using the point $ (q_s, p) = (460, 1.65) $:

\[
1.65 = 0.0003625 \cdot 460 + b_s
\]

\[
b_s = 1.65 - 0.16675 = 1.48325
\]

So the supply equation is:

\[
p = 0.0003625q_s + 1.48325
\]

### Step 3: Find the Market Equilibrium

The market equilibrium occurs where the quantity demanded equals the quantity supplied, $ q_d = q_s = q $, and the price $ p $ is the same for both demand and supply. Thus, set the demand equation equal to the supply equation:

\[
-0.0009q + 2.49 = 0.0003625q + 1.48325
\]

Combine like terms:

\[
2.49 - 1.48325 = 0.0009q + 0.0003625q
\]

\[
1.00675 = 0.0012625q
\]

Solve for $ q $:

\[
q = \frac{1.00675}{0.0012625} \approx 797.57
\]

### Step 4: Determine the Equilibrium Price

Substitute $ q $ back into either the demand or supply equation to find the equilibrium price $ p $. Using the demand equation:

\[
p = -0.0009(797.57) + 2.49 \approx 1.78
\]

### Final Answer:

The market equilibrium price is **$1.78** per loaf, and the equilibrium quantity is **798 loaves** (rounded to the nearest whole number).

Would you like further details on any part of the process? Here are some related questions:

1. How is the slope calculated in linear equations?
2. What does the market equilibrium represent in economics?
3. How can the intercept $ b $ be interpreted in this context?
4. What are the effects of a price ceiling on this market?
5. How would the equilibrium change if the demand or supply curve shifted?

**Tip:** In economics, understanding how to derive and interpret demand and supply curves is crucial for predicting market behavior.

Learn how to find market equilibrium price and quantity using demand and supply analysis with a detailed example from a bread company. Discover the intersection of demand and supply curves based on given price and quantity data points. Market equilibrium price: $1.78 per loaf. Equilibrium quantity: 798 loaves.

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