We have a grassland problem where grass grows uniformly. The field can feed 14 cows for 30 days, or 20 cows for 20 days. We need to find how many cows it can feed for 10 days. This is a classic problem involving both initial grass amount and continuous grass growth.
Let's set up our variables to solve this problem systematically. We have G₀ as the original amount of grass, r as the daily grass growth rate, and c as the daily consumption per cow. The key equation is that total available grass equals total consumption: original grass plus growth equals cows times consumption rate times days.
Now let's set up our equations from the given conditions. For 14 cows eating for 30 days, we get G₀ plus 30r equals 420c. For 20 cows eating for 20 days, we get G₀ plus 20r equals 400c. Subtracting the second equation from the first gives us 10r equals 20c, so r equals 2c. Substituting back, we find that G₀ equals 360c.
Now we can find our answer. We know that G₀ equals 360c and r equals 2c. For n cows eating for 10 days, we have the equation: G₀ plus 10r equals n times 10c. Substituting our known values: 360c plus 20c equals 10nc. This simplifies to 380c equals 10nc. Dividing both sides by 10c, we get n equals 38. Therefore, 38 cows can eat for 10 days.