Correct answer: C. ` (H - aN)(H - bN) < 0 `
Let `x` be the quantity of Product A and `y` be the quantity of Product B. The information provided can be expressed as a system of two linear equations:
1. `ax + by = H` (The total labor hours constraint)
2. `x + y = N` (The total unit production constraint)
The problem states that the company produces a positive quantity of both products, which means `x > 0` and `y > 0`.
To find the conditions on the parameters `a`, `b`, `H`, and `N`, we first solve the system for `x` and `y`. From equation (2), we can write `y = N - x`. Substituting this into equation (1):
`ax + b(N - x) = H`
`ax + bN - bx = H`
`x(a - b) = H - bN`
Since `a ≠ b`, we can divide by `(a - b)`:
`x = (H - bN) / (a - b)`
Now we find `y` by substituting the expression for `x` back into `y = N - x`:
`y = N - (H - bN) / (a - b)`
`y = (N(a - b) - (H - bN)) / (a - b)`
`y = (aN - bN - H + bN) / (a - b)`
`y = (aN - H) / (a - b)`
The conditions `x > 0` and `y > 0` lead to a system of two inequalities:
1. `(H - bN) / (a - b) > 0`
2. `(aN - H) / (a - b) > 0`, which is equivalent to `(H - aN) / (b - a) > 0`
Let's analyze these inequalities. There are two cases.
Case 1: `a > b`. In this case, `a - b` is positive. The inequalities become:
`H - bN > 0` => `H > bN`
`aN - H > 0` => `aN > H`
Combining these, we get `bN < H < aN`.
Case 2: `a < b`. In this case, `a - b` is negative. For the fractions to be positive, the numerators must be negative:
`H - bN < 0` => `H < bN`
`aN - H < 0` => `aN < H`
Combining these, we get `aN < H < bN`.
In both cases, `H` must be strictly between `aN` and `bN`. This condition, that a value (`H`) lies strictly between two other values (`aN` and `bN`), can be expressed compactly as the product of the differences being negative: `(H - aN)(H - bN) < 0`.
Choice A is incorrect because `(H - aN)(H - bN) > 0` implies that `H` is either greater than both `aN` and `bN` or less than both `aN` and `bN`. This would correspond to a situation requiring a negative quantity of one of the products.
Choice B is incorrect because `H = (a+b)N / 2` is the specific condition required to produce equal quantities of both products (`x = y = N/2`), but it is not a necessary condition for producing any positive, unequal mix of products.
Choice C is the correct answer. The condition `(H - aN)(H - bN) < 0` is a general way of stating that `H` must be between `aN` and `bN`, which is the necessary condition for both `x` and `y` to be positive.
Choice D is incorrect because `(H - aN)(H - bN) = 0` represents the boundary cases where `H = aN` or `H = bN`. In these cases, the quantity of one of the products would be exactly zero, which contradicts the requirement for a positive quantity of · both · products.