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:

$m=\frac{y_2-y_1}{x_2-x_1}$

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:

$m=\frac{39-24}{46-58}=\frac{15}{-12}=-\frac{5}{4}=-1.25$

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:

$P=-1.25q+96.5$