B3 A log in the shape of a cylinder is charred to the maximum possible brown with a square cross-section. What percentage is waste?

(7 points)
B3. We assume that the diagonal of the square is equal to the diameter of the circle. Thus, $r=\frac{d}{2}=\frac{a \sqrt{2}}{2}$. Therefore, $a=r \sqrt{2}$. The ratio of the volumes of the cube and the cylinder is $\frac{V_{k}}{V_{v}}=\frac{a^{2} v}{\pi r^{2} v}$. Considering that both bodies have the same height, we can simplify the fraction. We also substitute $a=r \sqrt{2}$ into the simplified fraction. We get $\frac{(r \sqrt{2})^{2}}{\pi r^{2}}=\frac{2}{\pi}=0.637$. Thus, $V_{k}=0.637 V_{v}$. Therefore, the volume of the cube represents $63.7\%$ of the volume of the cylinder. Hence, $36.3\%$ of the raspberries are wasted.

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![](https://cdn.mathpix.com/cropped/2024_06_07_c987701c9bb62194911fg-15.jpg?height=63&width=1687&top_left_y=777&top_left_x=230)

![](https://cdn.mathpix.com/cropped/2024_06_07_c987701c9bb62194911fg-15.jpg?height=63&width=1687&top_left_y=825&top_left_x=230)

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![](https://cdn.mathpix.com/cropped/2024_06_07_c987701c9bb62194911fg-15.jpg?height=68&width=1685&top_left_y=931&top_left_x=228)

Considering the same heights of the bodies (appropriate shortening) ..........................................................................

Written answer: Therefore, $36.3\%$ of the raspberries are wasted .....................................................