Join the growing community of curious minds on IDNLearn.com and get the answers you need. Ask your questions and get detailed, reliable answers from our community of experienced experts.

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.

Sagot :

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]