Solucionario Calculo Una Variable Thomas Finney Edicion 9 179 [ Web PLUS ]

[ V_{\max}= x^2 y = \Bigl(\frac{2R}{\sqrt{3}}\Bigr)^2 \cdot \frac{2R}{\sqrt{3}} = \frac{4R^2}{3} \cdot \frac{2R}{\sqrt{3}} = \frac{8R^3}{3\sqrt{3}}. ]

[ 4xR^2 - 3x^3 = 0 \quad\Longrightarrow\quad x\bigl(4R^2 - 3x^2\bigr) = 0. ] The professor nodded, his eyes twinkling

Setting the numerator to zero (the denominator never vanished inside the feasible interval) produced Well done, Maya

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.” A sphere centered at the origin

First, she rewrote the volume in a friendlier form for differentiation:

The vertices of the box lie on the sphere, so each corner satisfies the equation

She pulled a chair, settled into the worn leather, and spread out her notes. The room was quiet except for the distant hum of the campus heating system and the occasional rustle of a late‑night janitor’s cart. Maya began by sketching the situation on a scrap of graph paper. A sphere centered at the origin, radius R , and a rectangular box whose center coincided with the sphere’s center. Because the base was a square, she let x denote the length of one side of the base, and y the height of the box.