Which of the following is best modeled by a linear relationship that has a positive correlation?
The cost, y, in dollars, of manufacturing x items, where it costs $0.49 to manufacture each item.
This relationship is linear because the total cost increases at a constant rate of $0.49 for each additional item produced, reflecting a direct proportionality between the number of items and the total cost.
This scenario models exponential growth rather than a linear relationship, as the amount of money increases by a percentage each year, compounding over time. The growth rate accelerates, leading to a curve rather than a straight line.
This relationship also represents exponential growth, as the population of turtles grows by doubling each year. The increase is not constant; rather, it accelerates over time, which deviates from a linear correlation.
While the distance decreases linearly over time, it does not describe a positive correlation. Instead, it represents a negative relationship as the distance between the cyclists diminishes as they approach each other.
This situation represents a linear relationship as well, but it is characterized by a negative slope, signifying a descent rather than a positive correlation. The height decreases consistently over time.
The only choice that models a linear relationship with a positive correlation is the cost of manufacturing items, which increases steadily with each additional unit produced. Other options depict exponential growth or negative relationships, thus reinforcing that linear relationships must reflect constant change in the same direction.
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