1. The demand function of a product for a manufacturer is given as [tex]P(x) = ax + b[/tex]. The manufacturer knows that he can sell 1250 units when the price is GHS 5 per unit and he can sell 1500 units at a price of GHS 4 per unit. Find the total and average revenue functions.



Answer :

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]