To find the longest straw that can fit completely inside the juice can diagonally, we can use the Pythagorean theorem in three dimensions. The length of the diagonal (d) of a cylinder can be calculated with the formula:
\[ d = \sqrt{h^2 + r^2 + r^2} \]
Where:
- \( h \) is the height of the cylinder,
- \( r \) is the radius of the base of the cylinder.
First, we need to determine the radius \( r \) from the diameter. The diameter is given as 52 mm, so the radius is:
\[ r = \frac{diameter}{2} = \frac{52, mm}{2} = 26, mm \]
Substituting the values into the formula, we find the diagonal length:
\[ d = \sqrt{h^2 + 2r^2} \] \[ h = 122 , mm, \quad r = 26 , mm \]
Now substituting these values into the equation:
\[ d = \sqrt{122^2 + 2 \times 26^2} \]
Calculating \( 122^2 \):
\[ 122^2 = 14884 \]
Calculating \( 2 \times 26^2 \):
\[ 26^2 = 676 \quad \text{so} \quad 2 \times 676 = 1352 \]
Now, adding the two results:
\[ d = \sqrt{14884 + 1352} = \sqrt{16236} \]
Next, we calculate \( \sqrt{16236} \):
\[ \sqrt{16236} \approx 127.4 , mm \]
Thus, the longest straw that can fit completely inside the juice can diagonally is approximately:
\[ \boxed{127.4 , mm} \]