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 balloon’s 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.