Question: In a 19th century sea-battle, the number of ships on each side remaining t hours after the start are given by x(t) and y(t). If the ships are equally equipped, the relation between them is (x(t))2 -(y(t))2 = c, where c is a positive constant. The battle ends when one side has no ships remaining.
(a) If, at the start of the battle, 50 ships on one side oppose 40 ships on the other, what is the value of c? (b) If y(3) = 16, what is x(3)? What does this represent in terms of the battle?
(c) There is a time T when y(T )=0. What does this T represent in terms of the battle?
(d) At the end of the battle, how many ships remain on the victorious side?
(e) At any time during the battle, the rate per hour at which y loses ships is directly proportional to the number of x ships, with constant of proportionality k. Write an equation that represents this. Is k positive or negative? (f) Show that the rate per hour at which x loses ships is directly proportional to the number of y ships, with constant of proportionality k.
(g) Three hours after the start of the battle, x is losing ships at the rate of 32 ships per hour. What is k? At what rate is y losing ships at this time?