1. Establish rectangle dimensions: Assign length \( 2r \) and width \( h \), where \( 2r \) represents the cylinder base circumference and \( h \) its height. The perimeter equation is: \[ 2r + 2h = 300 \quad \Rightarrow \quad r + h = 150 \quad \Rightarrow \quad h = 150 - r. \] 2. Calculate cylinder volume: The volume formula is: \[ V = \pi r^2 h = \pi r^2 (150 - r). \] 3. Determine maximum \( V \): Calculate the derivative of \( V \) with respect to \( r \): \[ \frac{dV}{dr} = \pi \left[ 2r(150 - r) - r^2 \right] = \pi (300r - 3r^2). \] Set the derivative to zero: \[ 300r - 3r^2 = 0 \quad \Rightarrow \quad 3r(100 - r) = 0. \] Solutions are \( r = 0 \) or \( r = 100 \). Exclude \( r = 0 \) as it yields zero volume.
4. Apply second derivative test: The second derivative is: \[ \frac{d^2V}{dr^2} = \pi (300 - 6r). \] Evaluate at \( r = 100 \): \[ \frac{d^2V}{dr^2} = \pi (300 - 600) = -300\pi<0. \] This indicates that \( V \) reaches its maximum at \( r = 100 \).
5. Calculate \( h \): Substitute \( r \) back into the equation for \( h \): \[ h = 150 - r = 150 - 100 = 50. \]
Final Answer: The dimensions of the rectangular sheet are \( 2r = 200 \, {cm} \) and \( h = 50 \, {cm} \).