Math Problem Statement

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)?

Solution

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 NN (number of pies sold) as a function of the price PP (in dollars).

The problem gives two points:

  • When the price P=5P = 5, the demand N=100N = 100 pies.
  • When the price P=10P = 10, the demand N=55N = 55 pies.

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

To find the slope mm, use the formula for slope between two points (P1,N1)(P_1, N_1) and (P2,N2)(P_2, N_2):

m=N2N1P2P1=55100105=455=9m = \frac{N_2 - N_1}{P_2 - P_1} = \frac{55 - 100}{10 - 5} = \frac{-45}{5} = -9

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

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

N=9P+bN = -9P + b 100=9(5)+b100 = -9(5) + b 100=45+b100 = -45 + b b=145b = 145

Therefore, the demand function is:

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


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

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

slope=9 pies/dollar\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)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.

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Math Problem Analysis

Mathematical Concepts

Linear Equations
Slope-Intercept Form
Demand Functions

Formulas

N(P) = mP + b (Linear Equation for Demand)
m = (N2 - N1) / (P2 - P1) (Slope Calculation)

Theorems

Slope-Intercept Form
Linear Demand Models

Suitable Grade Level

Grades 9-11