To find the longest straw that can fit diagonally inside a cylindrical juice can, we can use the formula for the length of the diagonal (d) in a cylinder. The formula is derived from the Pythagorean theorem and is:
\[ d = \sqrt{h^2 + r^2 + r^2} \]
where \( h \) is the height of the cylinder and \( r \) is the radius of the base.
Given:
- Height \( h = 122 \) mm
- Diameter \( d = 52 \) mm, so the radius \( r = \frac{d}{2} = \frac{52}{2} = 26 \) mm
Now, we can calculate the diagonal:
\[ d = \sqrt{h^2 + 2r^2} \]
Substituting the values:
\[ d = \sqrt{122^2 + 2(26^2)} \]
Calculating \( 122^2 \) and \( 26^2 \):
\[ 122^2 = 14884 \] \[ 26^2 = 676 \]
Now plug these into the formula:
\[ d = \sqrt{14884 + 2 \times 676} \] \[ d = \sqrt{14884 + 1352} \] \[ d = \sqrt{16236} \]
Now calculate the square root:
\[ d \approx 127.4 \text{ mm} \]
Thus, the longest straw that can fit completely inside the juice can diagonally is approximately 127.4 mm.