A farming family raises cattle. The cattle graze tame pasture for 6 months and a

Question

A farming family raises cattle. The cattle graze tame pasture for 6 months and are fed hay for 6 months over winter. At the end of each fall, the family sells a portion of its herd to minimize overwintering costs. An average cow eats approximately 350Kg of dry forage per month and during the summer, the farm's grazing pastures have an average grass production of 1200Kg/ha. The farm has a total of 315 ha of grazing land and can produce a total of 400 bales, weighing an average of 750Kg, on their hayland. If the family wants to fully utilize their pastures' production, but maintain 50 bales of carry over, how many cattle should they graze and how many should they overwinter (i.e. how many cattle should they keep year round, and how many should they sell in the fall)?

Explanation / Answer

Let x be the number cows maintained during summer. Amount of grass produced = 1200*315 kg Since one cow eats 350kg per month, therefore amount eaten by cows = x*350*6 kg Since pastures production has to be fully utilized, therefore 350*6*x = 1200*315 => x = 1200*315/(350*6) = 180 Therefore 180 cows would be kept during summer. Now let y be the number of cows kept during winter. Bales produced = 400*750 kg Bales that has to be taken for carry over = 50 Therefore Bales that would be fed to cows = 350 Amount eaten by cows = y*350*6 kg y*350*6 = 350*750 => y = 750/6 = 125 Therefore 180 cows will be kept in summer and 125 cows during winter. i.e. 55 cows will be sold in winter.

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