the supply function on the market for a certain product takes the form: p(Q)=15+1.75Q. When the market price is p=50, there is an excess of quantity demanded over quantity supplied in the market of 80 units. at price p=15, the surplus of quantity demanded over quantity shown is 240 pieces. what total revenue do enterprises realize at the point of market equilibrium? (assume that the demand function is linear)

To solve this problem, we need to find the market equilibrium point where the quantity demanded equals the quantity supplied. This requires us to derive the demand function and solve for the equilibrium price and quantity. 

### Step-by-Step Solution:

1. **Supply Function:** Given that the supply function is:
   \[
   p(Q) = 15 + 1.75Q
   \]
   where \( p \) is the price and \( Q \) is the quantity supplied.

2. **Find the Demand Function:**  
   At price \( p = 50 \), there is an excess demand of 80 units. This means that the quantity demanded, \( Q_d \), is 80 units more than the quantity supplied, \( Q_s \).  
   At price \( p = 15 \), there is an excess supply of 240 units. This means that the quantity demanded, \( Q_d \), is 240 units less than the quantity supplied, \( Q_s \).

   Using the supply function, we can find \( Q_s \) for the two prices:

   - For \( p = 50 \):
     \[
     50 = 15 + 1.75Q_s \implies Q_s = \frac{50 - 15}{1.75} = 20
     \]

     Thus, \( Q_d = Q_s + 80 = 20 + 80 = 100 \).

   - For \( p = 15 \):
     \[
     15 = 15 + 1.75Q_s \implies Q_s = \frac{15 - 15}{1.75} = 0
     \]

     Thus, \( Q_d = Q_s - 240 = 0 - 240 = -240 \).

   From these two points \((Q_d, p)\), \((100, 50)\) and \((-240, 15)\), we can find the linear demand function in the form of \( p = a - bQ_d \).

3. **Calculate the Demand Function:**  
   Using the two points, we determine the slope \( b \) of the demand function:
   \[
   b = \frac{50 - 15}{100 - (-240)} = \frac{35}{340} = \frac{7}{68}
   \]

   Then the demand function can be written as:
   \[
   p = a - \frac{7}{68}Q_d
   \]

   To find \( a \), we substitute one of the points into the demand function. Using the point \((100, 50)\):
   \[
   50 = a - \frac{7}{68} \cdot 100 \implies 50 = a - \frac{700}{68}
   \]

   Solving for \( a \):
   \[
   a = 50 + \frac{700}{68} = 50 + 10.29 \approx 60.29
   \]

   Therefore, the demand function is:
   \[
   p = 60.29 - \frac{7}{68}Q
   \]

4. **Find the Equilibrium Point:**

   At equilibrium, quantity demanded equals quantity supplied:  
   \[
   15 + 1.75Q = 60.29 - \frac{7}{68}Q
   \]

   Combine like terms and solve for \( Q \):
   \[
   1.75Q + \frac{7}{68}Q = 60.29 - 15
   \]

   Convert and solve:
   \[
   1.75*+7=Revenue*=10,

Find_50_output

The supply function on the market for a certain product takes the form: p(Q)=15+1.75Q. When the market price is p=50, there is an excess of quantity demanded over quantity supplied in the market of 80 units. At price p=15, the surplus of quantity demanded over quantity shown is 240 pieces. What total revenue do enterprises realize at the point of market equilibrium? (Assume that the demand function is linear.)

Explore how to find the market equilibrium point using the supply and demand functions, with a detailed step-by-step solution. This problem involves key concepts such as linear equations and market equilibrium, suitable for students in Grades 10-12.

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