Two small rings each of weight w can slide on a fixed rough horizontal rod. the rings are attached one to each end of a uniform chain of weight 4w and length 4a so that the chain hangs freely from the rings. if the coefficient of friction between each ring and the rod is 1/2, find the distance between the rings when they are in limiting equilibrium. prove that the lowest point of the chain is then at a depth a below the rod.

The temperature in gavin's oven is a sinusoidal function of time. gavin sets his oven so that it has a maximum temperature of 280°f and a minimum temperature of 240°. once the temperature hits 280°, it takes 20 minutes before it is 280° again. gavin's cake needs to be in the oven for 30 minutes at temperatures at or above 270°. he puts the cake into the oven when it is at 260° and rising. how long will gavin need to leave the cake in the oven? (round your answer to the nearest minute.)