Complete the problem by using the accompanying figure, which shows a supply function and a demand function. (Assume price is measured in dollars.)

The q p-coordinate plane is given. There are 2 lines on the graph.
The first line enters the window at p = 10 on the positive p-axis, goes up and right, passes through the point (8, 14), passes through the point (30, 25) crossing the second line, passes through the point (36, 28), passes through the point (52, 36), and exits the window in the first quadrant.
The second line enters the window at p = 40 on the positive p-axis, goes down and right, passes through the point (8, 36), passes through the point (30, 25) crossing the first line, passes through the point (36, 22), passes through the point (52, 14), and exits the window in the first quadrant.
(a) If the price is $36, what quantity is demanded?
units

(b) If the price is $36, what quantity is supplied?
units

(c) Is there a surplus or shortage when the price is $36?
surplus
shortage

How many units is the surplus or shortage?
units

To solve this problem, we need to extract the supply and demand functions from the given data points and graph.

### (a) If the price is $36, what quantity is demanded?

We are asked to determine the quantity demanded when the price $ p = 36 $. To do this, we need the demand function, which can be deduced from the points provided.

From the description, the demand function has these points:
- $ (0, 40) $ (at price $ p = 40 $, quantity demanded is $ q = 0 $),
- $ (8, 36) $,
- $ (30, 25) $,
- $ (52, 14) $.

Let's determine the slope ($ m $) of the demand line using two points, say $ (0, 40) $ and $ (52, 14) $:
\[
m = \frac{14 - 40}{52 - 0} = \frac{-26}{52} = -\frac{1}{2}
\]
Now, use the point-slope form to find the demand equation. Using the point $ (0, 40) $:
\[
p - 40 = -\frac{1}{2}(q - 0)
\]
\[
p = 40 - \frac{1}{2}q
\]
This is the demand function:
\[
p = 40 - \frac{1}{2}q
\]

Now, solve for $ q $ when $ p = 36 $:
\[
36 = 40 - \frac{1}{2}q
\]
\[
\frac{1}{2}q = 40 - 36 = 4
\]
\[
q = 8
\]
Thus, the quantity demanded when the price is $ 36 $ is **8 units**.

### (b) If the price is $36, what quantity is supplied?

Now we need the supply function, which can be determined from the supply line’s points:
- $ (0, 10) $,
- $ (8, 14) $,
- $ (30, 25) $,
- $ (52, 36) $.

Let's determine the slope ($ m $) of the supply line using two points, say $ (0, 10) $ and $ (52, 36) $:
\[
m = \frac{36 - 10}{52 - 0} = \frac{26}{52} = \frac{1}{2}
\]
Now, use the point-slope form to find the supply equation. Using the point $ (0, 10) $:
\[
p - 10 = \frac{1}{2}(q - 0)
\]
\[
p = 10 + \frac{1}{2}q
\]
This is the supply function:
\[
p = 10 + \frac{1}{2}q
\]

Now, solve for $ q $ when $ p = 36 $:
\[
36 = 10 + \frac{1}{2}q
\]
\[
\frac{1}{2}q = 36 - 10 = 26
\]
\[
q = 52
\]
Thus, the quantity supplied when the price is $ 36 $ is **52 units**.

### (c) Is there a surplus or shortage when the price is $36?

At the price of $ 36 $, the quantity demanded is $ 8 $ units, and the quantity supplied is $ 52 $ units. Since the quantity supplied is greater than the quantity demanded, there is a **surplus**.

### (d) How many units is the surplus?

The surplus is the difference between the quantity supplied and the quantity demanded:
\[
\text{Surplus} = 52 - 8 = 44 \text{ units}
\]

Thus, there is a **surplus of 44 units** when the price is $ 36 $.

---

Would you like more details or have any questions? Here are 5 related questions to explore further:

1. How do supply and demand functions change with price adjustments?
2. What happens when the price is set at the equilibrium point?
3. Can you identify the equilibrium price and quantity from the given functions?
4. How do changes in external factors (like production costs) affect the supply function?
5. What would happen if the government imposed a price ceiling or floor?

**Tip:** The intersection of the supply and demand curves determines the equilibrium price and quantity, where supply equals demand.

Complete the problem by using the accompanying figure, which shows a supply function and a demand function. (Assume price is measured in dollars.)

The q p-coordinate plane is given. There are 2 lines on the graph.
The first line enters the window at p = 10 on the positive p-axis, goes up and right, passes through the point (8, 14), passes through the point (30, 25) crossing the second line, passes through the point (36, 28), passes through the point (52, 36), and exits the window in the first quadrant.
The second line enters the window at p = 40 on the positive p-axis, goes down and right, passes through the point (8, 36), passes through the point (30, 25) crossing the first line, passes through the point (36, 22), passes through the point (52, 14), and exits the window in the first quadrant.
(a) If the price is $36, what quantity is demanded?
(b) If the price is $36, what quantity is supplied?
(c) Is there a surplus or shortage when the price is $36? How many units is the surplus or shortage?

Learn how to analyze a supply and demand graph and calculate the surplus when the price is set at $36. This problem includes linear functions for supply and demand, covering key points such as calculating the slope and determining whether there is a surplus or shortage. Suitable for high school students in Grades 10-12.

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