3. Three friends enter Giorgio's pizzeria and each sit at one side of a rectangular table; the first is sitting at a side of length $70 \mathrm{~cm}$, the second and third sit opposite each other, on sides of length $l$. The pizzas have a diameter of $30 \mathrm{~cm}$; Giorgio serves the first customer's pizza so that it is tangent to his side of the table at the midpoint, and the pizzas of the other two so that they are tangent to their respective sides of the table and to the first pizza. What is the minimum value of $l$ (in centimeters) for which the three pizzas can fit entirely on the table?
(A) $10 \sqrt{5}$
(B) $20+5 \sqrt{5}$
(C) $15+10 \sqrt{5}$
(D) $30+10 \sqrt{5}$
(E) 60
3. The answer is (D). Consider a rectangular table where the length $l$ is the minimum such that the three pizzas are entirely contained on its surface, and that is, such that the second and third pizzas are tangent to the side at which none of the three friends are seated. Let $O_{1}, O_{2}$ be the centers of the first two pizzas, and let $P$ be the intersection point between the line through $O_{1}$ parallel to the side to which the second pizza is tangent (which has length $l$) and the line through $O_{2}$ parallel to the side to which the first pizza is tangent (which has length $70 \mathrm{~cm}$). Then $l=15+O_{1} P+15$; on the other hand, $O_{1} O_{2}$ is 60 $\mathrm{cm}$ long, while the length of $O_{2} P$ can be calculated, in centimeters, as $\frac{70}{2}-15=20$; the angle $O_{1} \widehat{P} O_{2}$ is right by definition, and thus by the Pythagorean Theorem $O_{1} P=\sqrt{O_{1} O_{2}{ }^{2}-O_{2} P^{2}}$, and so the desired length, in centimeters, is $30+10 \sqrt{5}$.