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      	     1.87     	    3.09     
      5.00      	  2.37  	  2.01   

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

We are provided with two data points for both the demand and supply functions, and we are tasked with determining the **quantity demanded (Qd)** when the price is \$8.90 per ton.

### Step 1: Find the demand equation.
A linear demand function can be written in the form:

\[
Q_d = m_d P + b_d
\]

Where:
- $ Q_d $ is the quantity demanded,
- $ P $ is the price,
- $ m_d $ is the slope of the demand function,
- $ b_d $ is the intercept of the demand function.

We are given two data points for demand:
- At $ P = 10.00 $, $ Q_d = 1.87 $,
- At $ P = 5.00 $, $ Q_d = 2.37 $.

We can calculate the slope $ m_d $ of the demand function using the formula for the slope between two points:

\[
m_d = \frac{Q_{d2} - Q_{d1}}{P_2 - P_1} = \frac{2.37 - 1.87}{5.00 - 10.00} = \frac{0.50}{-5.00} = -0.1
\]

Now, use one of the points to find the intercept $ b_d $. Let's use $ P = 10.00 $ and $ Q_d = 1.87 $:

\[
1.87 = (-0.1)(10.00) + b_d
\]
\[
1.87 = -1.00 + b_d
\]
\[
b_d = 2.87
\]

Thus, the demand function is:

\[
Q_d = -0.1P + 2.87
\]

### Step 2: Calculate the quantity demanded at $ P = 8.90 $.

Substitute $ P = 8.90 $ into the demand equation:

\[
Q_d = -0.1(8.90) + 2.87
\]
\[
Q_d = -0.89 + 2.87
\]
\[
Q_d = 1.9800 \, \text{tons}
\]

Thus, the quantity demanded when the price is \$8.90 per ton is **1.9800 tons**.

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Would you like more details or have any questions?

Here are 5 related questions for further practice:
1. How would you find the supply function for this product?
2. What is the equilibrium price and quantity for this market?
3. If the price increased to \$12.00, what would be the expected quantity demanded?
4. How does the slope of the demand curve affect market behavior?
5. How would you interpret the intercept of the demand function?

**Tip**: When solving linear equations, always check for both the slope and intercept carefully to ensure accuracy when applying the formula to new values.

A product has a linear demand function and a linear supply function. It is known that when the price of a product = P (in $/ton), the quantity demanded will equal to Qd (in tons), and the quantity supplied will equal to Qs (in tons). The detailed information is as follows:
P ($/ton)     Qd (tons)        Qs (tons) 
10.00         1.87             3.09 
5.00          2.37             2.01

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

Learn how to calculate the quantity demanded when the price is $8.90/ton using a linear demand function. This problem involves solving for the slope and intercept of the demand equation based on two price-quantity data points. Suitable for students in Grades 9-11.

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