In the equation \( y = 4x \), \( y \) represents the number of bales of hay needed per day and \( x \) represents the number of goats. The equation indicates that for every goat, there are 4 units of hay needed.
To determine the correct interpretation of the constant of proportionality, we can analyze the options given:
a. The constant of proportionality is four; there are four goats eating each bale of hay.
- Incorrect. This doesn't accurately reflect the relationship; rather, it suggests a misunderstanding of consumption.
b. The constant of proportionality is 4; each goat needs four bales of hay per day.
- Incorrect. This implies that one goat needs four bales, which is not the intended meaning of the equation.
c. The constant of proportionality is 4; every bale of hay feeds a goat.
- Incorrect. This suggests that each bale feeds only one goat, which is also a misunderstanding.
d. The constant of proportionality is 4; the farm needs four bales of hay per day.
- Partially correct but ambiguous. The statement could imply the farm as a whole needs a constant amount of hay, which does not depend on the number of goats.
However, none of the options fully explain the situation accurately. The best understanding of the constant of proportionality in this context is that it means for each goat, 4 units of hay are required. Hence, the closest correct interpretation among the provided options might be that the constant of proportionality refers to the ratio of hay needed per goat.
To summarize, the correct interpretation is that for every goat, 4 bales of hay are needed based on the equation \( y = 4x \) (where \( y \) is the total bales of hay and \( x \) is the number of goats). So it’s best summed up as: Each goat requires 4 bales of hay per day.
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