A product has a linear demand function and a linear supply function. It is known that when the price of a product = P (in $/ton), the quantity demanded will equals to Qd (in tons), and the quantity supplied will equals to Qs (in tons). The detailed information can be seen below.

P ($/ton)	     Qd (tons)     	     Qs (tons)      
     10.00      	     2.02     	    3.07     
      5.00      	  2.36  	  1.98   

Calculate the quantity demanded (tons) when the price is $8.9 /ton.
(Notes: Use 4 digits after decimal point)

To solve this problem, we first need to determine the linear demand and supply functions. We can represent the demand and supply functions as follows:

\[ Q_d = m_d P + b_d \]
\[ Q_s = m_s P + b_s \]

Where:
- $ Q_d $ is the quantity demanded,
- $ Q_s $ is the quantity supplied,
- $ P $ is the price,
- $ m_d $ and $ m_s $ are the slopes of the demand and supply functions respectively,
- $ b_d $ and $ b_s $ are the y-intercepts of the demand and supply functions respectively.

### Step 1: Determine the demand function ($ Q_d = m_d P + b_d $)

Using the provided data points for demand:
- When $ P = 10.00 $, $ Q_d = 2.02 $
- When $ P = 5.00 $, $ Q_d = 2.36 $

The slope of the demand function ($ m_d $) is calculated as:
\[
m_d = \frac{Q_{d2} - Q_{d1}}{P_2 - P_1} = \frac{2.36 - 2.02}{5.00 - 10.00} = \frac{0.34}{-5.00} = -0.068
\]

Now, we can find the y-intercept ($ b_d $) using one of the points (e.g., $ P = 10.00, Q_d = 2.02 $):
\[
Q_d = m_d P + b_d
\]
\[
2.02 = (-0.068)(10.00) + b_d
\]
\[
b_d = 2.02 + 0.68 = 2.70
\]

So, the demand function is:
\[
Q_d = -0.068 P + 2.70
\]

### Step 2: Calculate the quantity demanded when $ P = 8.9 $ using the demand function

Substitute $ P = 8.9 $ into the demand function:
\[
Q_d = -0.068(8.9) + 2.70
\]
\[
Q_d = -0.6052 + 2.70 = 2.0948 \, \text{tons}
\]

So, the quantity demanded when the price is $8.9/ton is **2.0948 tons**.

Would you like to see the details of the calculations or have any questions?

Here are 5 related questions you might find interesting:
1. How would the supply function be calculated given the data points?
2. What is the quantity supplied when the price is $8.9/ton?
3. How do demand and supply functions interact at equilibrium?
4. Can you calculate the equilibrium price and quantity using these functions?
5. How would changes in the slope of the demand curve affect the market?

**Tip:** Always check the units of your final answer to ensure consistency with the problem statement.

A product has a linear demand function and a linear supply function. It is known that when the price of a product = P (in $/ton), the quantity demanded will equal to Qd (in tons), and the quantity supplied will equal to Qs (in tons). Calculate the quantity demanded (tons) when the price is $8.9 /ton. Data points: P=10.00, Qd=2.02, Qs=3.07; P=5.00, Qd=2.36, Qs=1.98.

Learn how to calculate the quantity demanded when the price is $8.9/ton using linear demand and supply functions. This example explores key algebraic concepts, including slope calculation and linear equations, with specific data points for price and quantity demanded/supplied.

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