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To find the total and average revenue functions, we need to determine the constants [tex]\( a \)[/tex] and [tex]\( b \)[/tex] for the linear demand function [tex]\( P(x) = a x + b \)[/tex].
We have two known points on the demand curve:
1. At [tex]\( x = 1250 \)[/tex], [tex]\( P(x) = 5 \)[/tex]
2. At [tex]\( x = 1500 \)[/tex], [tex]\( P(x) = 4 \)[/tex]
Using these points, we can set up the following system of linear equations:
1. [tex]\( 5 = a \cdot 1250 + b \)[/tex]
2. [tex]\( 4 = a \cdot 1500 + b \)[/tex]
### Step 1: Find [tex]\( a \)[/tex]
Subtract the second equation from the first to eliminate [tex]\( b \)[/tex]:
[tex]\[ 5 - 4 = (a \cdot 1250 + b) - (a \cdot 1500 + b) \][/tex]
[tex]\[ 1 = a \cdot 1250 - a \cdot 1500 \][/tex]
[tex]\[ 1 = a (1250 - 1500) \][/tex]
[tex]\[ 1 = a (-250) \][/tex]
[tex]\[ a = -\frac{1}{250} \][/tex]
Therefore, [tex]\( a = -0.004 \)[/tex].
### Step 2: Find [tex]\( b \)[/tex]
Substitute [tex]\( a = -0.004 \)[/tex] back into one of the original equations to solve for [tex]\( b \)[/tex]. Using the first equation:
[tex]\[ 5 = (-0.004) \cdot 1250 + b \][/tex]
[tex]\[ 5 = -5 + b \][/tex]
[tex]\[ b = 10 \][/tex]
Thus, [tex]\( b = 10 \)[/tex].
So, the demand function is:
[tex]\[ P(x) = -0.004 x + 10 \][/tex]
### Step 3: Find the Total Revenue Function
The total revenue [tex]\( R(x) \)[/tex] is given by:
[tex]\[ R(x) = x \cdot P(x) \][/tex]
Substitute the demand function [tex]\( P(x) = -0.004 x + 10 \)[/tex]:
[tex]\[ R(x) = x \cdot (-0.004 x + 10) \][/tex]
[tex]\[ R(x) = -0.004 x^2 + 10 x \][/tex]
Therefore, the total revenue function is:
[tex]\[ R(x) = -0.004 x^2 + 10 x \][/tex]
### Step 4: Find the Average Revenue Function
The average revenue [tex]\( A(x) \)[/tex] is given by:
[tex]\[ A(x) = \frac{R(x)}{x} \][/tex]
Using the total revenue function:
[tex]\[ A(x) = \frac{-0.004 x^2 + 10 x}{x} \][/tex]
[tex]\[ A(x) = -0.004 x + 10 \][/tex]
Therefore, the average revenue function is:
[tex]\[ A(x) = -0.004 x + 10 \][/tex]
In summary, the total and average revenue functions are:
- Total Revenue Function: [tex]\( R(x) = -0.004 x^2 + 10 x \)[/tex]
- Average Revenue Function: [tex]\( A(x) = -0.004 x + 10 \)[/tex]
We have two known points on the demand curve:
1. At [tex]\( x = 1250 \)[/tex], [tex]\( P(x) = 5 \)[/tex]
2. At [tex]\( x = 1500 \)[/tex], [tex]\( P(x) = 4 \)[/tex]
Using these points, we can set up the following system of linear equations:
1. [tex]\( 5 = a \cdot 1250 + b \)[/tex]
2. [tex]\( 4 = a \cdot 1500 + b \)[/tex]
### Step 1: Find [tex]\( a \)[/tex]
Subtract the second equation from the first to eliminate [tex]\( b \)[/tex]:
[tex]\[ 5 - 4 = (a \cdot 1250 + b) - (a \cdot 1500 + b) \][/tex]
[tex]\[ 1 = a \cdot 1250 - a \cdot 1500 \][/tex]
[tex]\[ 1 = a (1250 - 1500) \][/tex]
[tex]\[ 1 = a (-250) \][/tex]
[tex]\[ a = -\frac{1}{250} \][/tex]
Therefore, [tex]\( a = -0.004 \)[/tex].
### Step 2: Find [tex]\( b \)[/tex]
Substitute [tex]\( a = -0.004 \)[/tex] back into one of the original equations to solve for [tex]\( b \)[/tex]. Using the first equation:
[tex]\[ 5 = (-0.004) \cdot 1250 + b \][/tex]
[tex]\[ 5 = -5 + b \][/tex]
[tex]\[ b = 10 \][/tex]
Thus, [tex]\( b = 10 \)[/tex].
So, the demand function is:
[tex]\[ P(x) = -0.004 x + 10 \][/tex]
### Step 3: Find the Total Revenue Function
The total revenue [tex]\( R(x) \)[/tex] is given by:
[tex]\[ R(x) = x \cdot P(x) \][/tex]
Substitute the demand function [tex]\( P(x) = -0.004 x + 10 \)[/tex]:
[tex]\[ R(x) = x \cdot (-0.004 x + 10) \][/tex]
[tex]\[ R(x) = -0.004 x^2 + 10 x \][/tex]
Therefore, the total revenue function is:
[tex]\[ R(x) = -0.004 x^2 + 10 x \][/tex]
### Step 4: Find the Average Revenue Function
The average revenue [tex]\( A(x) \)[/tex] is given by:
[tex]\[ A(x) = \frac{R(x)}{x} \][/tex]
Using the total revenue function:
[tex]\[ A(x) = \frac{-0.004 x^2 + 10 x}{x} \][/tex]
[tex]\[ A(x) = -0.004 x + 10 \][/tex]
Therefore, the average revenue function is:
[tex]\[ A(x) = -0.004 x + 10 \][/tex]
In summary, the total and average revenue functions are:
- Total Revenue Function: [tex]\( R(x) = -0.004 x^2 + 10 x \)[/tex]
- Average Revenue Function: [tex]\( A(x) = -0.004 x + 10 \)[/tex]
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