To find the longest straw that can fit completely inside a cylindrical juice can, we need to calculate the length of the diagonal that runs from the bottom center of the cylinder to the top edge of the cylinder.
To do this, we will use the Pythagorean theorem in three dimensions. We need to consider:
- The height of the cylinder.
- The radius of the cylinder (from its diameter).
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Calculate the radius of the cylinder: The diameter of the cylinder is 52 mm, so the radius \( r \) is half of that: \[ r = \frac{52 \text{ mm}}{2} = 26 \text{ mm} \]
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Cylinder height: The height \( h \) of the cylinder is given as 122 mm.
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Use the Pythagorean theorem in three dimensions: The longest straw corresponds to the diagonal of a rectangular prism formed by the radius and height of the cylinder. The diagonal \( d \) can be calculated using: \[ d = \sqrt{(2r)^2 + h^2} \] where \( 2r \) is the diameter (the width of the base) and \( h \) is the height of the cylinder.
Substituting the values we have: \[ d = \sqrt{(2 \times 26 \text{ mm})^2 + (122 \text{ mm})^2} = \sqrt{(52 \text{ mm})^2 + (122 \text{ mm})^2} \] \[ = \sqrt{2704 \text{ mm}^2 + 14884 \text{ mm}^2} \] \[ = \sqrt{17588 \text{ mm}^2} \] \[ \approx 132.6 \text{ mm} \]
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Round to the nearest tenth: \[ d \approx 132.6 \text{ mm} \quad (\text{rounding to the nearest tenth}) \]
Thus, the longest straw that can fit completely inside the juice can diagonally is approximately 132.6 mm.