3A. An isosceles trapezoid with height 12, leg 13, and midline 15, rotates around the shorter base. Calculate the volume of the resulting body.
Solution. From the conditions of the problem, we get $\frac{a+b}{2}=15, h=12, c=13$, where $a$ and $b$ are the lengths of the bases, $h$ is the height, and $c$ is the length of the leg of the trapezoid. From the Pythagorean Theorem, it follows that $\frac{a-b}{2}=\sqrt{13^{2}-12^{2}}=5$, from which we obtain $a=20$ and $b=10$.

The desired volume is $V=V_{1}-2 V_{2}$, where $V_{1}$ is the volume of a cylinder with base radius $h=12$ and height $a=20$, and $V_{2}$ is the volume of a cone with base radius $h=12$ and height $\frac{a-b}{2}=5$.

Thus, the desired volume is $V=12^{2} \pi \cdot 20-2 \cdot \frac{1}{3} 12^{2} \pi \cdot 5=2400 \pi$.

4AB. In a convex quadrilateral $A B C D$, $\overline{A D}=\overline{C D}$ and $\measuredangle A D C=90^{\circ}$. If $\overline{A B}=a, \overline{B C}=b, \overline{B D}=d$ and $\measuredangle A B C=\beta$, prove that $2 d^{2}=a^{2}+b^{2}+2 a b \sin \beta$.