What is the side length of the smallest square plate on which a \( 36-\mathrm{cm} \) chopstick can fit along a diagonal without any overhang? The approximate side length of the square plate is \( \square \mathrm{cm} \). (Round to the nearest tenth as needed.)

To determine the side length of the smallest square plate on which a \( 36-\mathrm{cm} \) chopstick can fit along a diagonal, we use the relationship between the diagonal \( d \) of a square and its side length \( s \). The formula for the diagonal of a square is given by: \[ d = s\sqrt{2} \] We need the diagonal to be at least \( 36 \mathrm{cm} \), so we set up the equation: \[ s\sqrt{2} = 36 \] To solve for \( s \), we divide both sides by \( \sqrt{2} \): \[ s = \frac{36}{\sqrt{2}} = \frac{36 \sqrt{2}}{2} = 18\sqrt{2} \] Now we calculate \( 18\sqrt{2} \). Using \( \sqrt{2} \approx 1.414 \): \[ s \approx 18 \times 1.414 \approx 25.452 \] Rounding to the nearest tenth, we get: \[ s \approx 25.5 \, \mathrm{cm} \] So, the approximate side length of the square plate is \( 25.5 \, \mathrm{cm} \).