To find the length of the longest straw that can fit diagonally inside the cylindrical juice can, we can visualize the problem as finding the length of a diagonal inside a cylinder. The longest diagonal will stretch from one point on the base of the cylinder to the opposite point on the top face.
The diagonal, \(d\), inside a cylinder can be calculated using the formula derived from the Pythagorean theorem for a right triangle formed by the height of the cylinder and the diameter of the cylinder's base:
\[ d = \sqrt{h^2 + r^2} \]
where:
- \(h\) is the height of the cylinder,
- \(r\) is the radius of the cylinder.
First, we need to find the radius of the base. Given the diameter is \(52\) mm, the radius \(r\) is:
\[ r = \frac{\text{diameter}}{2} = \frac{52 , \text{mm}}{2} = 26 , \text{mm} \]
Now we can substitute the values into the formula. The height \(h\) is given as \(122\) mm.
Now we compute the diagonal \(d\):
\[ d = \sqrt{h^2 + r^2} = \sqrt{122^2 + 26^2} \]
Calculating each squared term:
\[ 122^2 = 14884 \quad \text{and} \quad 26^2 = 676 \]
Adding these values together:
\[ 14884 + 676 = 15560 \]
Now, taking the square root to find \(d\):
\[ d = \sqrt{15560} \approx 124.0 , \text{mm} \]
Thus, the length of the longest straw that can fit completely inside the juice can diagonally is approximately:
\[ \boxed{124.0} , \text{mm} \]