Question

for each of the pairs of quantities, identify if they are directly, inversely, or not proportional. complete parts (a) through (j).
(a) tax ($) charged for buying an item : cost ($) of an item
choose the correct answer.
not proportional
directly proportional
inversely proportional
(b) population of a region : area of a region
directly proportional
not proportional
inversely proportional
(c) average speed to work : time it takes
not proportional
inversely proportional
directly proportional
(d) population of new york city : population of new york state
directly proportional
inversely proportional
not proportional

Explanation:

Part (a)

Tax is calculated as a percentage of the item's cost (e.g., sales tax: \( \text{Tax} = \text{Rate} \times \text{Cost} \)). As the cost of the item increases, the tax (at a constant rate) increases proportionally. So, tax and item cost are directly proportional.

Population of a region depends on many factors (density, resources, etc.), not just the area. A large area might have low population density, and a small area might have high density. There's no fixed proportional relationship between population and area.

The formula for distance is \( \text{Distance} = \text{Speed} \times \text{Time} \). If distance to work is constant, \( \text{Speed} = \frac{\text{Distance}}{\text{Time}} \). As speed increases, time decreases (and vice versa) for a fixed distance, so they are inversely proportional.

Answer:

directly proportional

Part (b)