Let $t$ be TNYWR.

One cylinder has a radius of $\sqrt{2}$ and a height of $a$. Another cylinder has a radius of $\sqrt{5}$ and a height of $b$. How many pairs of positive integers $(a, b)$ are there so that the sum of the volumes of the two cylinders is $10 \pi t$ ?
The volume of a cylinder with radius $\sqrt{2}$ and height $a$ is $\pi(\sqrt{2})^{2} a=2 \pi a$.

The volume of a cylinder with radius $\sqrt{5}$ and height $b$ is $\pi(\sqrt{5})^{2} b=5 \pi b$.

From the given information, $2 \pi a+5 \pi b=10 \pi t$ or $2 a+5 b=10 t$.

Since $a>0$ and $b>0$, then $2 a<10 t$ or $a<5 t$. Also, $5 b<10 t$ or $b<2 t$.

Rearranging $2 a+5 b=10 t$ gives $2 a=10 t-5 b$ or $a=5 t-\frac{5}{2} b$.

If we assume that $t$ is a positive integer, then for $a$ to be a positive integer, it must be the case that $b$ is even.

Since $b<2 t$, then the possible even values of $b$ less than $2 t$ are $2,4,6, \ldots, 2 t-4,2 t-2$. There are $t-1$ such values.

Under the assumption that $t$ is a positive integer, these are exactly the values of $b$ that give positive integer values for $a$.

In particular, we obtain the pairs $(a, b)=(5 t-5,2),(5 t-10,4), \ldots,(10,2 t-4),(5,2 t-2)$, of which there are $t-1$.

Since the answer to (b) is 30 , then $t=30$ (which is a positive integer), and so $t-1=29$. That is, there are 29 pairs of positive integers $(a, b)$ that satisfy the requirements.

ANSWER: $6,30,29$