See Slow Cows

(a) More Slow Cows.

If 14 cows take exactly 5 times as long to graze out a pasture as 54 cows, and 38 cows take exactly 5 times as long as 174 cows, what is the largest prime number of cows that can graze the same pasture?

Note 1: In this part, all times are calculated in exact whole numbers of days, and the pasture is of the smallest size compliant with the given conditions.

(b) Newton's Goats

Six goats eat all the grass in a pasture in exactly three days, while 3 goats take exactly seven days to graze out the same pasture. On what day will the pasture become exhausted, if two goats are grazing there?

Note 2: In both parts, the grass grows at a steady rate, and each animal eats the same, constant, amount each day; but, unlike cows, goats eat grass down to the stubble, so that areas that have already been grazed do not resume growth while the pasture remains in use.

Broll,

Using trial and error method I arrived at following conclusion.

If x is the rate at which the goats eat and y is the rate of growth of grass left uneaten and c is the initial grass in pasture then

c=16.60807x and y=.086286x. This assumes that the grass doesnt grow while the goats are eating. In this case the pasture is free of grass on the 14th day by a very small margin. i.e. pasture is only containing 0.24% of original grass on 14th day. For all practical purpose the pasture is free of grass on the 13th day.