Math Problem Statement
Help me solve the equations
Solution
To solve the questions in the image, let’s break it down step by step.
Part (a): Complete the Table
The problem gives us:
- The base admission price is , with attendees.
- For every $1 increase in the admission price, the park loses 250 attendees. This means:
- The revenue is calculated as .
Step 1: Complete the missing values for and
For :
For :
The completed table is:
Admission Price () | Attendance () | Revenue () |
---|---|---|
$10 | 25000 | $250,000 |
$11 | 24750 | $272,250 |
$12 | 24500 | $294,000 |
Part (b): Write the formulas
Formula for :
As derived earlier, the formula for attendance is: Simplify:
Formula for :
The revenue is the price times the number of attendees: Substitute into this: Simplify:
So the revenue formula is:
Part (c): Maximum Revenue
To find the price that maximizes revenue, we use the vertex formula for a quadratic equation. The formula for the vertex (maximum point) is: Here, and :
Thus, the admission price that produces the maximum revenue is .
Maximum Revenue:
Substitute into :
The maximum revenue is , and the corresponding attendance is:
Final Results
-
Completed Table:
Admission Price () Attendance () Revenue () $10 25000 $250,000 $11 24750 $272,250 $12 24500 $294,000 -
Attendance formula:
-
Revenue formula:
-
Maximum revenue occurs at , with and .
Let me know if you need further clarification or calculations! Here are five related questions for practice:
- How does the attendance change when the price increases by $1?
- What is the attendance when the price is $30?
- What is the revenue when the price is $20?
- Can the park achieve revenue higher than $800,000? Why or why not?
- Why is the revenue a quadratic function of the price?
Tip: Always check for the domain of your price values when working with quadratic revenue equations!
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Math Problem Analysis
Mathematical Concepts
Linear Relationships
Quadratic Functions
Revenue Maximization
Formulas
N(p) = 27500 - 250p
R(p) = -250p^2 + 27500p
Vertex formula for quadratic functions: p = -b / 2a
Theorems
Vertex theorem for maximum or minimum of a quadratic function
Suitable Grade Level
Grades 9-12
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