There is exactly one isosceles triangle that has a side of length 10 and a side of length 22 . What is the perimeter of this triangle?
An isosceles triangle must have at least two sides of equal length.

If an isosceles triangle has at least one side of length 10 and at least one side of length 22, then its three side lengths are either 10, 10, and 22 or 10,22 , and 22 .

In any triangle, the sum of the lengths of any two sides must be larger than the length of the other side.

Since $10+10<22$, a triangle cannot have sides of length 10 , 10, and 22 .

Therefore, the triangle's sides have lengths 10,22 , and 22 , so the answer is $10+22+22=54$.

ANSWER: 54