A balloon carrying an x-ray telescope and other scientific equipment must be designed and launched. A rough measure of performance can be expressed in terms of the height reached by the balloon and the weight of the equipment lifted. Clearly, the height itself is a function of the balloons volume.

From past experience, it has been concluded that a satisfactory performance function to be maximized is P = f (V, W) = 100V 0.3V ² + 80W 0.2W² where V is the volume, and W the equipment weight.

The project to be undertaken has a budget constraint of $1040. The cost associated with the volume V is 2V, and the cost of the equipment is 4W. In order to ensure that a reasonable balance is obtained between performance due to the height and that due to the scientific equipment, the designer has to meet the constraint 80W 100V.

Find the optimal design in terms of volume and equipment weight, solving by the Frank-Wolfe algorithm.