justins father says that he will not plant less than a square meter of any vegetable.design a garden with the largest possible amount of land planted in potatoes that fits the condition of the problem and hat least one square meter allotted for each vegetable.
To design a garden with the largest possible amount of land planted in potatoes while fitting the condition stated, we can use a mathematical technique called optimization.
First, let's assume we have x square meters of land planted with potatoes. Since each vegetable must have at least one square meter allotted, the remaining land would be (100 - x) square meters.
To maximize the land planted in potatoes, we need to maximize the value of x. However, there is a constraint that each vegetable must have at least one square meter allotted. This means (100 - x) square meters must be divided among the other vegetables.
Since we want to maximize the land planted in potatoes, we can assume that all the remaining land should be divided equally among the other vegetables. This means each of the remaining vegetables would have (100 - x) / n square meters allocated, where n is the number of vegetables (excluding potatoes).
To satisfy the condition, we must have (100 - x) / n >= 1, which simplifies to (100 - x) >= n.
To find the largest possible value of x, we need to find the largest possible value of n that satisfies the inequality (100 - x) >= n. The largest possible value of n can be found by subtracting 1 from (100 - x).
Therefore, the largest possible amount of land planted in potatoes is x = (100 - (n+1)) square meters.
Now, let's solve a numerical example:
Assuming there are 4 vegetables (excluding potatoes), we can maximize the land planted in potatoes by letting n = 4. Therefore, the largest possible amount of land planted in potatoes would be:
x = (100 - (4+1))
x = 95 square meters.
Thus, in this case, we can design a garden with 95 square meters planted in potatoes, while ensuring at least one square meter is allotted for each vegetable.