To find the longest straw that can fit completely inside the cylindrical juice can diagonally, we can use the formula for the diagonal (d) of a cylinder, which is given by:
\[ d = \sqrt{h^2 + r^2 + r^2} \]
where \( h \) is the height of the cylinder and \( r \) is the radius. The radius \( r \) is half of the diameter.
First, we need to find the radius:
\[ r = \frac{diameter}{2} = \frac{52 \text{ mm}}{2} = 26 \text{ mm} \]
Now we have:
- Height, \( h = 122 \) mm
- Radius, \( r = 26 \) mm
Next, we calculate the length of the diagonal:
\[ d = \sqrt{h^2 + 2r^2} \] \[ d = \sqrt{(122)^2 + 2(26)^2} \]
Calculating \( (122)^2 \):
\[ (122)^2 = 14884 \]
Calculating \( (26)^2 \):
\[ (26)^2 = 676 \]
Now substituting the values into the equation for \( d \):
\[ d = \sqrt{14884 + 2(676)} \] \[ d = \sqrt{14884 + 1352} \] \[ d = \sqrt{16236} \]
Now we calculate \( \sqrt{16236} \):
\[ \sqrt{16236} \approx 127.4 \]
Thus, the longest straw that can fit completely inside the juice can diagonally, rounded to the nearest tenth, is:
\[ \boxed{127.4 \text{ mm}} \]