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:

$N(P) = -9P + 145$

(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.