Of course! Let's break down the problem step by step.
### Given Demand Function:
[tex]\[ p = 10,000 - 2Q \][/tex]
### Quantities Sold:
- [tex]\( Q_1 = 500 \)[/tex]
- [tex]\( Q_2 = 2500 \)[/tex]
We need to calculate the following:
1. The price for each quantity ([tex]\( p_1 \)[/tex] and [tex]\( p_2 \)[/tex])
2. The Total Revenue (TR) for each quantity
3. The Marginal Revenue (MR)
### Step 1: Calculate the Price for Each Quantity
For [tex]\( Q_1 = 500 \)[/tex]:
[tex]\[ p_1 = 10,000 - 2 \times 500 \][/tex]
[tex]\[ p_1 = 10,000 - 1,000 \][/tex]
[tex]\[ p_1 = 9,000 \][/tex]
For [tex]\( Q_2 = 2500 \)[/tex]:
[tex]\[ p_2 = 10,000 - 2 \times 2500 \][/tex]
[tex]\[ p_2 = 10,000 - 5,000 \][/tex]
[tex]\[ p_2 = 5,000 \][/tex]
### Step 2: Calculate the Total Revenue (TR) for Each Quantity
Total Revenue (TR) is calculated as Price (p) times Quantity (Q).
For [tex]\( Q_1 = 500 \)[/tex]:
[tex]\[ TR_1 = p_1 \times Q_1 \][/tex]
[tex]\[ TR_1 = 9,000 \times 500 \][/tex]
[tex]\[ TR_1 = 4,500,000 \][/tex]
For [tex]\( Q_2 = 2500 \)[/tex]:
[tex]\[ TR_2 = p_2 \times Q_2 \][/tex]
[tex]\[ TR_2 = 5,000 \times 2500 \][/tex]
[tex]\[ TR_2 = 12,500,000 \][/tex]
### Step 3: Calculate the Marginal Revenue (MR)
Marginal Revenue (MR) is the change in Total Revenue divided by the change in Quantity.
[tex]\[ \Delta TR = TR_2 - TR_1 \][/tex]
[tex]\[ \Delta TR = 12,500,000 - 4,500,000 \][/tex]
[tex]\[ \Delta TR = 8,000,000 \][/tex]
[tex]\[ \Delta Q = Q_2 - Q_1 \][/tex]
[tex]\[ \Delta Q = 2500 - 500 \][/tex]
[tex]\[ \Delta Q = 2,000 \][/tex]
[tex]\[ MR = \frac{\Delta TR}{\Delta Q} \][/tex]
[tex]\[ MR = \frac{8,000,000}{2,000} \][/tex]
[tex]\[ MR = 4,000 \][/tex]
### Summary of Results:
- [tex]\( p_1 = 9,000 \)[/tex]
- [tex]\( TR_1 = 4,500,000 \)[/tex]
- [tex]\( p_2 = 5,000 \)[/tex]
- [tex]\( TR_2 = 12,500,000 \)[/tex]
- [tex]\( MR = 4,000 \)[/tex]
Hence, we have calculated the Total Revenue and Marginal Revenue for the given quantities.
