Question

Find the volume of the largest cube that can be cut from a whose a log of cross section whose radius is 12 centimeters

Answer 1

Consider a square inscribed in a circle. The radius of a wood log or a circle is equal to $1/2$ of its diagonal $d.$ By Pythagorean Theorem, find the side of the square $a$

$$a^2+a^2=d^2$$

$$a=d/\sqrt{2}$$

Substitute

$$a=\dfrac{2(12\ cm)}{\sqrt{2}}=12\sqrt{2}\ cm$$

If the length of the log is no less than $12\sqrt{2}\ cm,$ then the volume of the largest cube that can be cut from a whose a log is

$$V=a^3=(12\sqrt{2}\ cm)^3=3456\sqrt{2}\ cm^3$$