$6\mathrm{.}14$ Two heat exchangers in a circulating water loop transfer heat from a fluid condensing at $175F$ to a boiling fluid at $68F,$ as shown in Figure P$6\mathrm{.}14.$ The overall heat transfer coefficients of the evaporator and condenser are $U_E=5\mathrm{.}3Bt\frac{u}{h}*f{t}^2*F$ and $U_C=7Bt\frac{u}{h}*f{t}^{2}F,$ respectively. The first cost of the heat exchangers is $$\frac{21\mathrm{.}50}{f}{t}^2,$ and the present worth of the lifetime pumping costs is $11\mathrm{.}75[$$$*\frac{h}{l}bm]m_w^{}.$ This system must accomplish a heat transfer rate of $220,000Bt\frac{u}{h}$ between the condensing and boiling fluids. Using a numerical optimization algorithm, determine the heat exchanger areas, $A_1$ and $A_2,$ that minimize the total present worth of costs.

FIGURE P$6\mathrm{.}14$