Question

The consumers will demand 46 units when the price of a product is $39, and $58 units when the price is $24. Find the demand function (express the price) P in terms of the quantity q), assuming it is linear. P = ________

Answer 1

Given that:

- The consumers will demand 46 units when the price of a product is $39.

- The consumers will demand $58 units when the price is $24.

1. You need to remember the Slope-Intercept Form of the equation of a line:

`y=mx+b`

Where "m" is the slope of the line and "b" is the y-intercept.

2. In order to find the slope of the line, you can use this formula:

`m=(y_2-y_1)/(x_2-x_1)`

Where these two points are on the line:

`(x_1,y_1);(x_2,y_2)_{}`

In this case, knowing that:

`\begin{gathered} x=q \\ y=P \end{gathered}`

Then, you can identify these two points:

`(46,39);(58,24)`

Therefore, you can set up that:

`\begin{gathered} y_2=39 \\ y_1=24 \\ x_2=46 \\ x_1=58 \end{gathered}`

Hence, substituting values into the formula, you get:

`m=(39-24)/(46-58)=(15)/(-12)=-(5)/(4)=-1.25`

3. In order to find the value of "b", you can follow these steps:

- Substitute one of the points and the slope, into this equation:

`P=mq+b`

Then:

`39=(-1.25)(46)+b`

- Solve for "b":

`\begin{gathered} 39=(-1.25)(46)+b \\ \\ 39=-57.5+b \\ \\ 39+57.5=b \\ \\ b=96.5 \end{gathered}`

4. Knowing the slope and the y-intercept, you can write the following Demand Function:

`P=-1.25q+96.5`

Hence, the answer is:

`P=-1.25q+96.5`