https://file.aiquickdraw.com/mathbot/file/17302522515500ook6kkp.jpg

To solve this, let’s analyze the problem step-by-step.

### Problem Breakdown
1. A rectangle is constructed such that:
   - Its base lies on the x-axis.
   - The two upper vertices are on the parabola given by \( y = 4 - x^2 \).

2. We are tasked with finding the **shorter dimension** of this rectangle.

### Solution Steps
1. **Determine the Coordinates of the Upper Vertices:**
   Since the upper vertices lie on the parabola \( y = 4 - x^2 \), assume the rectangle’s vertices on the parabola are at points \( (x, y) \) and \( (-x, y) \), with \( y = 4 - x^2 \).

2. **Dimensions of the Rectangle:**
   - **Width (Horizontal dimension):** The distance between the points \( (x, y) \) and \( (-x, y) \) is \( 2x \).
   - **Height (Vertical dimension):** The height is simply the \( y \)-coordinate, which is \( 4 - x^2 \).

3. **Determine Which Dimension is Shorter:**
   To find the shorter dimension, we need to analyze the values of \( 2x \) and \( 4 - x^2 \) and determine the conditions under which one is smaller than the other.

4. **Optimizing the Shorter Dimension:**
   The shorter dimension will depend on the range of \( x \) values that make either \( 2x \) or \( 4 - x^2 \) the minimum value. Since \( x \) can range between 0 and 2 (because beyond \( x = 2 \), \( y \) would be negative, which doesn’t make sense in this context), we can analyze the function behavior in this interval.

Would you like the detailed calculations, including the function behavior analysis to confirm which dimension is shorter?

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Here are some questions to further explore related concepts:

1. How do we find the maximum area of the rectangle formed within the parabola?
2. What would change if the parabola equation was different, say \( y = x^2 - 4 \)?
3. Can we find an equation to generalize the shorter dimension for any quadratic function \( y = a - bx^2 \)?
4. How would this problem change if the base of the rectangle was not on the x-axis?
5. How does the symmetry of the parabola affect the rectangle's dimensions?

**Tip:** For problems involving shapes within curves, consider symmetry and boundary conditions to simplify calculations.

A rectangle is constructed with its base on the x-axis and its upper two vertices on the parabola y = 4 - x^2. The shorter dimension of the rectangle is

Explore how to determine the shorter dimension of a rectangle with its base on the x-axis and upper vertices on the parabola y = 4 - x^2. This problem uses key concepts in geometry and optimization, ideal for students in Grades 10-12.

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