She realized that the story of Exercise 179 wasn’t just about finding a maximum volume. It was about translating a three‑dimensional picture into algebra, about the elegance of a single variable governing a whole family of shapes, and about the quiet satisfaction that comes from turning a “hard problem” into a “solved puzzle”.
Using the product rule and the chain rule, she obtained
Finally, the maximal volume:
She felt a surge of satisfaction. The problem had been reduced to a single‑variable function, exactly as the title promised. The next step was to find the maximum of (V(x)). Maya knew she needed the derivative (V'(x)) and the critical points where it vanished (or where the derivative was undefined). She set her mind to the task.
Factoring out the common denominator gave She realized that the story of Exercise 179
Simplifying gave
[ 3x^2 = 4R^2 \quad\Longrightarrow\quad x = \frac{2R}{\sqrt{3}}. ] The problem had been reduced to a single‑variable
As she walked home, she imagined the inscribed cube—edges perfectly aligned, each corner just touching the sphere—sitting like a gem inside a glass sphere, a concrete reminder that sometimes, the most beautiful solutions are the simplest, and that every calculus problem hides a story waiting to be told.
[ V(x) = 2x^2 \bigl(R^2 - \tfrac{x^2}{2}\bigr)^{1/2}. ] She set her mind to the task
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