$1.$ A tech company produces two types of gadgets: SmartWatches and FitnessBands. They aim to maximize their weekly profit considering the availability of components and production capacity. SmartWatch sells for a profit of $$40$ each, and FitnessBand for $$20$ each.

There are two key components in producing their gadgets: microchips and sensors$.$ Each Smart$-$ Watch requires $3$ microchips and $2$ sensors$,$ while each FitnessBand requires $1$ microchip and $3$ sensors$.$ The company has $400$ microchips and $600$ sensors available per week. Also, the company wants to ensure that at least $50\%$ of the total gadgets are SmartWatches due to their popularity.

a$.$ Formulate the problem as a linear program.

b$.$ Solve the linear program from part $($a$)$ above using Excel. Specifically, find the optimal num$-$ ber of each gadget type and the optimal profit of the tech company. Please, submit a screenshot of your Excel spreadsheet including the solver parameters, the answer report, and the sensitivity report.

c$.$ If the profit from each SmartWatch increases to $$50,$ will this affect the optimal number of SmartWatch? What is the new maximum profit?