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(A) Let the radius of the base be $R$ and the height be $H$. The surface area $A$ of an open cylinder is given by $A = 2\pi RH + \pi R^2$. From this,w

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Height of the cylinder

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(A) Let the radius of the base be $R$ and the height be $H$. The surface area $A$ of an open cylinder is given by $A = 2\pi RH + \pi R^2$. From this,we can express $H$ as $H = \frac{A - \pi R^2}{2\pi R}$. The volume $V$ is given by $V = \pi R^2 H = \pi R^2 \left( \frac{A - \pi R^2}{2\pi R} \right) = \frac{R}{2}(A - \pi R^2) = \frac{AR}{2} - \frac{\pi R^3}{2}$. To find the maximum volume,we differentiate $V$ with respect to $R$: $\frac{dV}{dR} = \frac{A}{2} - \frac{3\pi R^2}{2}$. Setting $\frac{dV}{dR} = 0$,we get $A = 3\pi R^2$. Checking the second derivative,$\frac{d^2V}{dR^2} = -3\pi R$,which is negative for $R > 0$,confirming a maximum. Substituting $A = 3\pi R^2$ into the surface area formula: $3\pi R^2 = 2\pi RH + \pi R^2$. This simplifies to $2\pi R^2 = 2\pi RH$,which implies $R = H$. Thus,the radius is equal to the height of the cylinder.

If an open cylinder of given surface area has maximum volume,then its radius is

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