Problem 19 (Optimization)Find all pairs of positive integer numbersx, ysuch thatx+ 4y= 100and theirproduct is the largest possible.And:We have two variablesx, ysatisfyingx= 100-4y. We want to maximize the functionP(x, y) =xy)P(y) = (100-4y)y= 100y-4y2in the interval [0,25] (ify >25 thenx <0). We getP0(y) = 100-8y)P0(y) = 0,y=252We have a critical point aty= 12.5. In particular,y <12.5)P0(y)>0andy >12.5)P0(y)<0ThereforePattains the absolute maximum aty= 12.5 (moreover it’s obvious thatP(0) =P(25) = 0 at theendpoints). The maximum corresponds to the pair (x, y) = (50,12.5).Since we are interested in integer solutions, we should look at the closest integers and we obtain the pairs:(x, y) = (52,12))xy= 624and(x, y) = (48,13))xy= 624Problem 20 (Optimization)A metal storage tank with volumeVis to be constructed in the shape of a rightcircular cylinder surmounted by a hemisphere. What dimensions will require the least amount of metal?