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1. Consider the following diagram of the demand curve of a good for an individual taken from page 105 of the textbook.

(a)   Find the equation of the demand curve above.

(b)  Find the quantity demanded for the individual when the price of the good is $2.5. (c) What are the total benefits of the individual if she consumes 11 units of the good?

Let's solve this problem step-by-step.

### (a) Equation of the Demand Curve:
The graph shows a linear demand curve. To find the equation of the line, we need to use the slope-intercept form of a line:

\[
P = mQ + b
\]

Where:
- $ P $ is the price on the y-axis (in dollars),
- $ Q $ is the quantity on the x-axis (in units),
- $ m $ is the slope of the demand curve,
- $ b $ is the y-intercept (where the curve crosses the y-axis).

**Step 1: Identify two points from the graph:**
From the graph, two clear points are:
- Point 1: $ (Q = 0, P = 4) $ — This is the y-intercept where the curve touches the price axis.
- Point 2: $ (Q = 20, P = 0) $ — This is the x-intercept where the curve touches the quantity axis.

**Step 2: Calculate the slope:**
The slope $ m $ is calculated as:

\[
m = \frac{{P_2 - P_1}}{{Q_2 - Q_1}} = \frac{0 - 4}{20 - 0} = -\frac{4}{20} = -\frac{1}{5}
\]

**Step 3: Write the equation of the line:**
Now, using the slope-intercept form and the y-intercept $ b = 4 $, the equation of the demand curve is:

\[
P = -\frac{1}{5}Q + 4
\]

### (b) Quantity demanded when the price is $2.5:
To find the quantity demanded when the price $ P = 2.5 $, substitute $ P = 2.5 $ into the demand curve equation:

\[
2.5 = -\frac{1}{5}Q + 4
\]

Now, solve for $ Q $:

\[
2.5 - 4 = -\frac{1}{5}Q
\]
\[
-1.5 = -\frac{1}{5}Q
\]
\[
Q = \frac{1.5 \times 5}{1} = 7.5
\]

So, the quantity demanded when the price is $2.5 is **7.5 units**.

### (c) Total benefits when 11 units are consumed:
Total benefits (or consumer surplus) can be found by calculating the area under the demand curve up to 11 units. This area is a triangle formed between the price axis, the demand curve, and the vertical line at $ Q = 11 $.

**Step 1: Find the price at $ Q = 11 $:**
Substitute $ Q = 11 $ into the demand curve equation:

\[
P = -\frac{1}{5}(11) + 4 = -\frac{11}{5} + 4 = 2.8
\]

So, the price when $ Q = 11 $ is $2.8.

**Step 2: Calculate the area under the curve:**
The total benefits are the area of the triangle with:
- Base = $ Q = 11 $ units,
- Height = Price when $ Q = 0 $ (which is $4).

The area $ A $ of the triangle is:

\[
A = \frac{1}{2} \times \text{Base} \times \text{Height} = \frac{1}{2} \times 11 \times 4 = 22
\]

Thus, the total benefits for consuming 11 units is **$22**.

### Summary:
- (a) The equation of the demand curve is $ P = -\frac{1}{5}Q + 4 $.
- (b) The quantity demanded when the price is $2.5 is **7.5 units**.
- (c) The total benefits of consuming 11 units is **$22**.

Would you like more details on any part of the solution?

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Here are 5 related questions to explore further:

1. What happens to the total benefits if the price drops to $1?
2. How would the demand curve change if the maximum price consumers are willing to pay increases to $5?
3. What is the elasticity of demand at $ Q = 11 $?
4. How would a subsidy of $1 per unit affect the quantity demanded?
5. How does consumer surplus change when the price is $3?

**Tip**: When calculating areas under demand curves, remember that triangles represent consumer surplus when dealing with linear demand curves!

1. Consider the following diagram of the demand curve of a good for an individual taken from page 105 of the textbook.

(a) Find the equation of the demand curve above.

(b) Find the quantity demanded for the individual when the price of the good is $2.5.
(c) What are the total benefits of the individual if she consumes 11 units of the good?

Learn how to find the equation of a demand curve, determine the quantity demanded at a specific price, and calculate the total benefits from consuming a certain quantity of goods. This problem involves key algebraic concepts and is suitable for students in Grades 9-12.

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