To find the total revenue (TR) and marginal revenue (MR) when the quantity (Q) is 2, we follow a step-by-step approach:
### Step 1: Calculate the Average Revenue (AR)
The average revenue (AR) function is given by:
[tex]\[ AR = 20 - 4Q \][/tex]
Substituting [tex]\( Q = 2 \)[/tex]:
[tex]\[ AR = 20 - 4 \times 2 \][/tex]
[tex]\[ AR = 20 - 8 \][/tex]
[tex]\[ AR = 12 \][/tex]
### Step 2: Calculate the Total Revenue (TR)
Total revenue (TR) is obtained by multiplying the average revenue (AR) by the quantity (Q):
[tex]\[ TR = AR \times Q \][/tex]
Using the values we have:
[tex]\[ TR = 12 \times 2 \][/tex]
[tex]\[ TR = 24 \][/tex]
### Step 3: Calculate the Marginal Revenue (MR)
Marginal revenue (MR) is the derivative of the total revenue (TR) with respect to the quantity (Q). First, we express TR in terms of Q:
[tex]\[ AR = 20 - 4Q \][/tex]
Therefore,
[tex]\[ TR = Q \times AR \][/tex]
[tex]\[ TR = Q \times (20 - 4Q) \][/tex]
[tex]\[ TR = 20Q - 4Q^2 \][/tex]
To find MR, we take the derivative of TR with respect to Q:
[tex]\[ MR = \frac{d(TR)}{dQ} = \frac{d}{dQ}(20Q - 4Q^2) \][/tex]
[tex]\[ MR = 20 - 8Q \][/tex]
Substituting [tex]\( Q = 2 \)[/tex] into the MR function:
[tex]\[ MR = 20 - 8 \times 2 \][/tex]
[tex]\[ MR = 20 - 16 \][/tex]
[tex]\[ MR = 4 \][/tex]
### Summary
At [tex]\( Q = 2 \)[/tex]:
- The average revenue (AR) is [tex]\( 12 \)[/tex].
- The total revenue (TR) is [tex]\( 24 \)[/tex].
- The marginal revenue (MR) is [tex]\( 4 \)[/tex].
