Question

Minimum Surface Area Use Lagrange multipliers to find the dimensions of a right circular cylinder with volume \(V_{0}\) cubic units and minimum surface area.

Short answer

The solution will be the values of \(r\) (radius), \(h\) (height) and \(\lambda\) (the Lagrange multiplier) obtained by solving the equations derived from the differentiation of the Lagrange function.

Step-by-step solution

Formulate Equations

To start, the formulation of equations for the volume and the surface area of the cylinder is required. They are given by \(V = \pi r^2 h\) and \(A = 2 \pi r h + 2 \pi r^2\) where \(r\) is the radius, \(h\) is the height, and \(\pi\) denotes the constant Pi

Apply the Constraint

The volume \(V\) of the cylinder is fixed as \(V_{0}\), therefore, we can write the equation \( \pi r^2h = V_{0}\) or \(h = V_{0}/(\pi r^2)\)

Substitute the Constraint into the Area Equation

Substitute the value of \(h\) obtained in step 2 into the formula of surface area, the equation now becomes \(A = 2 \pi r (V_{0}/\(\pi r^2\)) + 2 \pi r^2\) or \(A = 2V_{0}/r + 2\pi r^2\)

Set Up the Lagrange Function

Now, set the Lagrange function \(L = A - \lambda(V - V_{0}) = 2V_{0}/r + 2\pi r^2 - \lambda(\pi r^2 h - V_{0})\)

Differentiate and Set to Zero

Differentiate Lagrange function with respect to \(r\), \(h\) and \(\lambda\), and set them each to zero to find the necessary conditions for \(L\) to reach its minimum value

Solve the Equations

Solving these three equations will provide the values of \(r\), \(h\) and \(\lambda\) that minimize the surface area of the cylinder