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The question shown in the image provides information about the demand for pies as a function of price. Let's break it down and solve it step by step.

### (a) Determine the demand $ N $ (number of pies sold) as a function of the price $ P $ (in dollars).

The problem gives two points:
- When the price $ P = 5 $, the demand $ N = 100 $ pies.
- When the price $ P = 10 $, the demand $ N = 55 $ pies.

We can use the formula for a line $ N(P) = mP + b $, where $ m $ is the slope and $ b $ is the intercept.

To find the slope $ m $, use the formula for slope between two points $ (P_1, N_1) $ and $ (P_2, N_2) $:

\[
m = \frac{N_2 - N_1}{P_2 - P_1} = \frac{55 - 100}{10 - 5} = \frac{-45}{5} = -9
\]

So, the slope is $ m = -9 $.

Now, substitute $ m = -9 $ and use one of the points to solve for $ b $. Let's use $ P = 5 $ and $ N = 100 $:

\[
N = -9P + b
\]
\[
100 = -9(5) + b
\]
\[
100 = -45 + b
\]
\[
b = 145
\]

Therefore, the demand function is:

\[
N(P) = -9P + 145
\]

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### (b) What is the slope of the graph of $ N(P) $?

From the previous calculation, the slope of the graph is:

\[
\text{slope} = -9 \text{ pies/dollar}
\]

The slope represents the rate of change in the number of pies sold for each dollar increase in price. Specifically, for each 1-dollar increase in price, 9 fewer pies are sold.

---

If you have any questions or need more details, feel free to ask! 

### Related Questions:
1. How would the graph of $ N(P) $ look like in terms of its intercepts?
2. What happens to the demand when the price reaches $0?
3. How would the demand change if the price increased by $2?
4. What is the significance of the y-intercept in this context?
5. How does elasticity relate to this linear demand function?

### Tip:
Whenever dealing with a linear equation, the slope will always give you important insights into how one variable changes in response to another. In economics, it’s often used to understand demand or supply sensitivity.

Suppose the demand for Colin's kidney pies is linear in the price P. Further, assume that he can sell 100 pies when the price is $5.00 and 55 pies when the price is $10.00. (a) Determine the demand N (number of pies sold) as a function of the price P (in dollars). (b) What is the slope of the graph of N(P)?

Learn how to calculate the demand function for pies based on a linear model where price and demand are given. This detailed step-by-step solution covers slope calculations and demand functions. Suitable for high school students in Grades 9-11.

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