Suppose that the supply function for honey is p=​S(q)=0.4q+2.8​, where p is the price in dollars for an 8​-oz container and q is the quantity in barrels. Suppose also that the equilibrium price is ​$4.80 and the demand is 4 barrels when the price is ​$6.20. Find an equation for the demand​ function, assuming it is linear.

To find the equation for the demand function, we'll assume that it is linear and has the form:

\[
p = D(q) = mq + b
\]

where $ p $ is the price, $ q $ is the quantity in barrels, $ m $ is the slope of the demand curve, and $ b $ is the y-intercept.

### Step 1: Use the information about the demand when the price is $6.20.
When $ p = 6.20 $, $ q = 4 $ barrels. So, we have the equation:
\[
6.20 = 4m + b \quad \text{(Equation 1)}
\]

### Step 2: Use the information about the equilibrium price.
At the equilibrium price of $ p = 4.80 $, supply equals demand. We know that the supply function is $ S(q) = 0.4q + 2.8 $.

Set $ S(q) = 4.80 $ and solve for $ q $:
\[
4.80 = 0.4q + 2.8
\]
\[
4.80 - 2.8 = 0.4q
\]
\[
2 = 0.4q
\]
\[
q = 5
\]

So, at the equilibrium price $ p = 4.80 $, the quantity $ q $ is 5 barrels. Therefore, the demand function at equilibrium can also be written as:
\[
4.80 = 5m + b \quad \text{(Equation 2)}
\]

### Step 3: Solve the system of linear equations.
Now, we'll solve the system of equations formed by Equation 1 and Equation 2:
\[
6.20 = 4m + b
\]
\[
4.80 = 5m + b
\]

Subtract Equation 2 from Equation 1:
\[
(6.20 - 4.80) = (4m + b) - (5m + b)
\]
\[
1.40 = -m
\]
\[
m = -1.40
\]

### Step 4: Find the y-intercept $ b $.
Substitute $ m = -1.40 $ back into Equation 1:
\[
6.20 = 4(-1.40) + b
\]
\[
6.20 = -5.60 + b
\]
\[
b = 6.20 + 5.60
\]
\[
b = 11.80
\]

### Final Demand Function
The equation for the demand function is:
\[
p = -1.40q + 11.80
\]

This is the linear demand function.

Would you like more details or have any questions?

### Related Questions:
1. How do we interpret the slope of the demand function?
2. What would happen to the demand if the price increased by $1?
3. How does the equilibrium price relate to supply and demand curves?
4. What is the price elasticity of demand given this function?
5. How would changes in supply affect the equilibrium price?

### Tip:
Remember, in a linear demand function, a negative slope indicates that as price increases, the quantity demanded decreases, which is typical for most goods.

Learn how to derive a linear demand function using supply and equilibrium price data. This problem involves calculating the slope and intercept of a demand curve based on given price and quantity data.

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