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 (12, 16), passes through the point (30, 25) crossing the second line, passes through the point (34, 27), passes through the point (48, 34), 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 (12, 34), passes through the point (30, 25) crossing the first line, passes through the point (34, 23), passes through the point (48, 16), and exits the window in the first quadrant.
(a) If the price is $34, what quantity is demanded?
units

(b) If the price is $34, what quantity is supplied?
units

(c) Is there a surplus or shortage when the price is $34?
surplus
shortage

How many units is the surplus or shortage?
units

The problem provides two functions, a **demand function** and a **supply function**, represented by lines on a $ q $-$ p $ coordinate plane, where $ p $ represents price and $ q $ represents quantity. We need to analyze this information and determine the answers based on the graph described.

### (a) If the price is $34, what quantity is demanded?
From the problem description, the **demand line** passes through the points (12, 34) and (48, 16), with price $ p $ and quantity $ q $. Let's first derive the demand equation using these two points.

To find the slope $ m $ of the demand line:
\[
m = \frac{p_2 - p_1}{q_2 - q_1} = \frac{16 - 34}{48 - 12} = \frac{-18}{36} = -\frac{1}{2}
\]
Now, using the point-slope form of the line equation:
\[
p - p_1 = m(q - q_1)
\]
Substitute one of the points, say (12, 34):
\[
p - 34 = -\frac{1}{2}(q - 12)
\]
Simplify this equation:
\[
p = -\frac{1}{2}(q - 12) + 34
\]
Distribute the slope:
\[
p = -\frac{1}{2}q + 6 + 34
\]
\[
p = -\frac{1}{2}q + 40
\]
This is the **demand function**: $ p = -\frac{1}{2}q + 40 $.

Now, set $ p = 34 $ to find the corresponding quantity demanded:
\[
34 = -\frac{1}{2}q + 40
\]
Solving for $ q $:
\[
34 - 40 = -\frac{1}{2}q
\]
\[
-6 = -\frac{1}{2}q
\]
\[
q = 12
\]
Thus, the **quantity demanded** at $ p = 34 $ is **12 units**.

### (b) If the price is $34, what quantity is supplied?
The **supply line** passes through the points (12, 16) and (48, 34). Let's derive the supply equation using these points.

The slope of the supply line:
\[
m = \frac{34 - 16}{48 - 12} = \frac{18}{36} = \frac{1}{2}
\]
Using the point-slope form again, with the point (12, 16):
\[
p - 16 = \frac{1}{2}(q - 12)
\]
Simplify this equation:
\[
p = \frac{1}{2}(q - 12) + 16
\]
Distribute the slope:
\[
p = \frac{1}{2}q - 6 + 16
\]
\[
p = \frac{1}{2}q + 10
\]
This is the **supply function**: $ p = \frac{1}{2}q + 10 $.

Now, set $ p = 34 $ to find the corresponding quantity supplied:
\[
34 = \frac{1}{2}q + 10
\]
Solving for $ q $:
\[
34 - 10 = \frac{1}{2}q
\]
\[
24 = \frac{1}{2}q
\]
\[
q = 48
\]
Thus, the **quantity supplied** at $ p = 34 $ is **48 units**.

### (c) Is there a surplus or shortage when the price is $34?

When the price is $ 34 $, the quantity demanded is $ 12 $ units, and the quantity supplied is $ 48 $ units. Since the supply exceeds the demand, there is a **surplus**.

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

The surplus is the difference between the quantity supplied and the quantity demanded:
\[
\text{Surplus} = 48 - 12 = 36 \text{ units}.
\]

### Final Answers:
- (a) Quantity demanded at $ p = 34 $ is **12 units**.
- (b) Quantity supplied at $ p = 34 $ is **48 units**.
- (c) There is a **surplus**.
- (d) The surplus is **36 units**.

---

Do you need more details on any part of this solution? Here are some related questions to consider:

1. What is the equilibrium price where the supply equals the demand?
2. How would the graph shift if the price of production decreased?
3. What happens to the surplus if the price increases to $40?
4. Can you calculate the exact coordinates where the supply and demand curves intersect?
5. How does the slope of a line relate to the rate of change in supply or demand?

**Tip**: Remember that supply and demand curves are linear representations of more complex economic behaviors in many cases, but can be a good approximation at small scales!

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 (12, 16), passes through the point (30, 25) crossing the second line, passes through the point (34, 27), passes through the point (48, 34), 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 (12, 34), passes through the point (30, 25) crossing the first line, passes through the point (34, 23), passes through the point (48, 16), and exits the window in the first quadrant. (a) If the price is $34, what quantity is demanded? units (b) If the price is $34, what quantity is supplied? units (c) Is there a surplus or shortage when the price is $34? surplus shortage How many units is the surplus or shortage? units

This problem involves analyzing supply and demand functions to find the quantity demanded and supplied at a price of $34. Includes a detailed breakdown of linear equations, the point-slope form, and identifying surplus or shortage in the market. Suitable for students in Grades 9-12.

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