To find the longest straw that can fit inside a cylindrical juice can diagonally, we can use the three-dimensional distance formula, which involves the height, diameter, and the length of the diagonal across the base of the cylinder.
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Determine the radius of the cylinder.
The radius \( r \) is half of the diameter: \[ r = \frac{d}{2} = \frac{52 \text{ mm}}{2} = 26 \text{ mm} \] -
Identify the height of the cylinder. The height \( h \) is given as 122 mm.
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Calculate the diagonal length.
The diagonal \( D \) inside the cylinder can be found using the Pythagorean theorem in three dimensions: \[ D = \sqrt{h^2 + (2r)^2} \] where \( 2r \) is the diameter of the base. -
Substitute the values: \[ D = \sqrt{(122)^2 + (52)^2} \] First, calculate \( (122)^2 \) and \( (52)^2 \): \[ (122)^2 = 14884 \] \[ (52)^2 = 2704 \]
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Add these values: \[ D = \sqrt{14884 + 2704} = \sqrt{17588} \]
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Calculate the square root: \[ D \approx 132.6 \text{ mm} \]
Rounding to the nearest tenth, the longest straw that can fit completely inside the juice can diagonally is: \[ \boxed{132.6} \text{ mm} \]