Math Problem Statement

Consumers will demand 1365 items at a price of 2.75 per item. When the price per item increases to 4, the number of items demanded will decrease by 75 find the linear demand function p(x) for this item in slope intercept form

Solution

To find the linear demand function p(x)p(x), where pp represents the price and xx represents the number of items demanded, we will use the slope-intercept form of a line:

p(x)=mx+bp(x) = mx + b

where:

  • mm is the slope,
  • bb is the y-intercept.

Step 1: Identify the known points.

We have two points based on the problem statement:

  • At x=1365x = 1365 items, p=2.75p = 2.75 dollars (first point).
  • At x=136575=1290x = 1365 - 75 = 1290 items, p=4p = 4 dollars (second point).

Thus, the points are (1365,2.75)(1365, 2.75) and (1290,4)(1290, 4).

Step 2: Find the slope mm.

The formula for the slope between two points is:

m=p2p1x2x1m = \frac{p_2 - p_1}{x_2 - x_1}

Substitute the values from the two points:

m=42.7512901365=1.2575=160m = \frac{4 - 2.75}{1290 - 1365} = \frac{1.25}{-75} = -\frac{1}{60}

So, the slope m=160m = -\frac{1}{60}.

Step 3: Use the slope-intercept form to find bb.

Now, use one of the points (say (1365,2.75)(1365, 2.75)) and the slope m=160m = -\frac{1}{60} to find the y-intercept bb. Plug into the equation p=mx+bp = mx + b:

2.75=160(1365)+b2.75 = -\frac{1}{60}(1365) + b

Simplify:

2.75=22.75+b2.75 = -22.75 + b

b=2.75+22.75=25.5b = 2.75 + 22.75 = 25.5

Step 4: Write the final demand function.

The linear demand function is:

p(x)=160x+25.5p(x) = -\frac{1}{60}x + 25.5

This is the linear demand function in slope-intercept form.


Would you like further details or have any questions? Here are some related questions to consider:

  1. How does the slope mm represent the relationship between price and demand?
  2. What does the y-intercept bb tell us in this context?
  3. How would the function change if the demand decreased

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Math Problem Analysis

Mathematical Concepts

Algebra
Linear Equations
Slope-Intercept Form

Formulas

Slope formula: m = (p2 - p1) / (x2 - x1)
Slope-intercept form: p(x) = mx + b

Theorems

Linear Function Theory

Suitable Grade Level

Grades 9-12