https://r2.erweima.ai/mathbot/image/17270155893125v2p632s.pdf

The document appears to present a math exercise, specifically a problem involving geometry and optimization.

Here’s a breakdown of the problem:

- **Triangle**: The triangle \( ABC \) is isosceles, with \( I \) as the midpoint of side \( BC \). Given that \( BC = 10 \) and \( AI = 15 \), this forms a geometric base.
- **Rectangle MNQP**: A rectangle is inscribed in this setup, with \( BP = x \), where \( x \in [0,5] \).
- The problem asks to:
  1. Show that the area of the rectangle \( MNQP \) is given by the function: 
     \[
     f(x) = -6x^2 + 30x
     \]
  2. Provide a detailed table showing the variations of the function \( f \).
  3. a. Determine the value of \( x \) that maximizes the area of the rectangle and compute the maximum area.
     b. Solve for the values of \( x \) such that the area of \( MNQP \) is at least one-third of the area of the triangle \( ABC \).

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Would you like a step-by-step explanation of the solutions for each part?

Here are 5 relative questions for you:
1. How do you derive the area function \( f(x) = -6x^2 + 30x \)?
2. What is the process to create a variation table for a quadratic function?
3. How do you find the maximum area using the vertex form of a quadratic function?
4. What is the significance of comparing the area of the rectangle to the area of triangle \( ABC \)?
5. How can the inequality \( f(x) \geq \text{one-third of the triangle's area} \) be solved?

**Tip**: The vertex of a parabola \( ax^2 + bx + c \) provides the maximum (or minimum) of the quadratic function, and can be found using \( x = -\frac{b}{2a} \).

Given an isosceles triangle ABC with BC = 10 and AI = 15, a rectangle MNQP is inscribed. BP = x where x belongs to [0,5]. The user is asked to: 1. Show that the area of MNQP is f(x) = -6x^2 + 30x. 2. Provide the variation table for f(x). 3. Find the value of x that maximizes the area and solve for when the area is at least one-third of the triangle's area.

Solve a geometry optimization problem involving an isosceles triangle ABC and an inscribed rectangle MNQP. Learn how to find the quadratic area function, its maximum, and compare the area to the triangle's area. Suitable for students in Grades 10-12.

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