### Problem StatementAn open-top cylindrical container is to have a volume of \(216 \, \text{cm}^3\). What dimensions (radius and height) will minimize the surface area? ---This problem invites students to explore the relationship between volume and surface area in the context of an open-top cylinder. The objective is to determine the optimal dimensions, specifically the radius and height, that will result in the smallest possible surface area while maintaining a fixed volume.### Additional Notes- Volume of a cylinder, \( V \), is given by: \[ V = \pi r^2 h \] where \( r \) is the radius and \( h \) is the height.- Surface area of an open-top cylinder, \( A \), is given by: \[ A = \pi r^2 + 2\pi rh \]To solve this optimization problem, you'll typically use concepts from calculus, such as taking derivative(s), to find the minimum surface area given the constraint on volume.

The objective of this question is to find the dimensions (radius and height) of an open-top cylindrical container that will minimize the surface area, given that the volume of the container is 216 cubic centimeters.First, let's define the variables and equations. Let's denote the radius of the cylinder as r (in centimeters) and the height as h (in centimeters). The volume V of a cylinder is given by the formula V = πr^2h and the surface area A of an open-top cylinder is given by the formula A = πr^2 + 2πrh.Given that the volume of the cylinder is 216 cubic centimeters, we can substitute this into the volume formula and solve for h: 216 = πr^2h. Simplifying this equation gives us h = 216 / (πr^2).Next, we substitute h from the volume equation into the surface area equation: A = πr^2 + 2πr(216 / (πr^2)). Simplifying this equation gives us A = πr^2 + 432/r.To find the dimensions that minimize the surface area, we need to find the derivative of the surface area equation with respect to r, set it equal to zero, and solve for r. The derivative of A with respect to r is dA/dr = 2πr - 432/r^2.Setting the derivative equal to zero gives us 2πr - 432/r^2 = 0. Multiplying through by r^2 gives us 2πr^3 - 432 = 0. Solving this equation for r gives us r = (432 / (2π))^(1/3) = 3.919 cm (rounded to three decimal places).Substituting r = 3.919 cm into the equation for h gives us h = 216 / (π(3.919)^2) = 5.535 cm (rounded to three decimal places).The dimensions that will minimize the surface area of an open-top cylindrical container with a volume of 216 cubic centimeters are a radius of approximately 3.919 centimeters and a height of approximately 5.535 centimeters.