Math Problem Statement

A rectangle is inscribed between the graphs of y = 4 - x^2 and y = (1/4)x^4 - 1. Find the width of the rectangle that has the largest area.

Solution

The problem describes a rectangle inscribed between two curves:

  1. y=4−x2y = 4 - x^2
  2. y=14x4−1y = \frac{1}{4}x^4 - 1

The goal is to find the width of the rectangle that has the largest area.

To solve this, we need to:

  1. Express the area of the rectangle in terms of a variable.
  2. Use calculus to find the value of that variable that maximizes the area.

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Math Problem Analysis

Mathematical Concepts

Calculus
Optimization
Functions
Area of a Rectangle

Formulas

Area of a rectangle A = length * width

Theorems

First Derivative Test for Maximum

Suitable Grade Level

Grades 11-12

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