Question 1

The cost of manufacturing items 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, represents a direct linear relationship. As the number of items produced increases, the total cost increases proportionally, resulting in a positive correlation.

A) The amount of money, y, in dollars, in a bank account x years after the initial deposit, where the money in the account earns 9% interest per year and there are no other transactions in the account

This option describes exponential growth rather than a linear relationship. The amount of money increases at a rate that compounds over time due to interest, leading to a curve rather than a straight line, thus negating a positive correlation.

B) The cost, y, in dollars, of manufacturing x items, where it costs $0.49 to manufacture each item

This option correctly describes a linear relationship. The cost increases by a constant amount ($0.49) for each additional item produced, illustrating a direct and positive correlation between the number of items and the total cost.

C) The number of turtles, y, in a lake x years after introducing them to a lake, where the number of turtles doubles each year

This relationship is exponential, as the population of turtles grows by a factor of two each year. Consequently, it does not represent a linear relationship, and thus does not show a simple positive correlation.

D) The distance, y, in miles, between two cyclists traveling toward each other x hours after they both start moving, where each cyclist travels at a constant rate of 12 miles per hour

While this option does depict movement towards each other at a constant rate, the relationship between the time and the distance decreases as they approach each other, resulting in a non-linear correlation. Therefore, it does not represent a positive correlation.

E) The height, y, in feet, of a balloon x minutes after it begins descending, where the balloon descends at a constant rate of 2 feet per minute

This describes a linear relationship, but it represents a negative correlation as the height decreases over time. Thus, it does not fit the criteria of a positive correlation.

Conclusion

Option B is definitively correct as it exemplifies a linear relationship with a constant rate of cost increase per item manufactured, leading to a positive correlation. All other options either demonstrate exponential growth, negative correlations, or do not maintain a linear relationship, which disqualifies them from being the best model for a positive correlation.