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# Optimization Problem?

A rectangle is inscribed between two parabolas y=4x^2 and y=30-x^2. What is the area of the largest rectangle that could be inscribed?

Please Help! I have no clue.

### 1 Answer

- δοτζοLv 71 decade agoFavorite Answer
Find where they intersect to find the bounds on the width

4x² = 30 - x²

5x² = 30

x² = 6

x = ±√6

So the width must be between 0 and 2√6 (exclusive) and is given by the function 2x. The height of the rectangle is given by the difference of the functions, larger minus smaller.

h = 30 - x² - 4x² = 30 - 5x²

A = hw = (30-5x²)(2x) = 60x - 10x³

A' = 60 - 30x² = 0 ⇒ x = ±√2 ⇒ x = √2

w = 2√2

h = 20

A = 40√2 ≈ 56.569

The key to this whole thing is graphing the 2 equations and trying to visualize an inscribed rectangle.