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 , where represents the price and represents the number of items demanded, we will use the slope-intercept form of a line:
where:
- is the slope,
- is the y-intercept.
Step 1: Identify the known points.
We have two points based on the problem statement:
- At items, dollars (first point).
- At items, dollars (second point).
Thus, the points are and .
Step 2: Find the slope .
The formula for the slope between two points is:
Substitute the values from the two points:
So, the slope .
Step 3: Use the slope-intercept form to find .
Now, use one of the points (say ) and the slope to find the y-intercept . Plug into the equation :
Simplify:
Step 4: Write the final demand function.
The linear demand function is:
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:
- How does the slope represent the relationship between price and demand?
- What does the y-intercept tell us in this context?
- 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