Plugging this back into the expression for :
Maya solved for in terms of x :
After the class, several classmates gathered around Maya, peppering her with questions. She explained how the symmetry of the sphere forced the optimal box to be a cube, and how the derivative’s denominator reminded her to stay within the physically meaningful interval (0 < x < \sqrt{2},R). Later that night, Maya returned the Thomas & Finney volume to its shelf, the thin solution sheet now neatly folded back into place. She closed the library’s heavy door and stepped into the cool campus air, the bell of the clock tower echoing the rhythm of her thoughts. Plugging this back into the expression for :
[ y = 2\sqrt{R^2 - \frac{1}{2}\Bigl(\frac{2R}{\sqrt{3}}\Bigr)^2} = 2\sqrt{R^2 - \frac{1}{2}\cdot\frac{4R^2}{3}} = 2\sqrt{R^2 - \frac{2R^2}{3}} = 2\sqrt{\frac{R^2}{3}} = \frac{2R}{\sqrt{3}}. ] She closed the library’s heavy door and stepped
A ripple of impressed murmurs ran through the class. The professor nodded, his eyes twinkling. “Excellent,” he said. “You’ve illustrated perfectly how a multivariable problem can sometimes be reduced to one variable, and how the critical point tells us the shape of the optimal object. Well done, Maya.” The professor nodded, his eyes twinkling
Maya wrote the result in bold, underlined it, and added a small smiley face next to it—her personal signature of triumph. The next morning, the professor walked into the seminar room, a stack of papers in his hand. He asked the class to volunteer a solution for Exercise 179. Maya’s hand rose, heart thudding like a metronome.