1. The resistance to bending of a beam with a rectangular cross-section is proportional to the product of the width of this section and the square of its height. What should be the width of the cross-section of a beam cut from a round log with a diameter of $15 \sqrt{3}$ so that its resistance to bending is maximized?

(5 points)
# Solution.

$F(a)=k a b^{2}, a-$ width of the beam, $b$ - height of the beam. Since $b^{2}=225 \cdot 3-a^{2}$, then

$F(a)=k a\left(225 \cdot 3-a^{2}\right)=k\left(225 \cdot 3 a-a^{3}\right), F^{\prime}(a)=k\left(225 \cdot 3-3 a^{2}\right)=3 k\left(225-a^{2}\right)$.

The derivative equals zero and changes sign from positive to negative at $a=15$, at which point the resistance to bending will be the greatest.

Answer: 15.