For each of the relationships below, explain whether you think it is best described by a linear function or a non-linear function. Explain your reasoning.
a. A person's height as a function of the person's age (from age 0 to 100)
b. The probability of getting into a car accident as a function of the speed at which you drive
c. The time it takes you to get to work as a function the speed at which you drive
a. A person's height as a function of the person's age: This relationship is typically described by a non-linear function. Initially, as a person grows from infancy to adolescence, their height increases rapidly. However, during adulthood, the rate of increase in height slows down, and eventually, it levels off. Therefore, a linear function would not accurately capture this relationship since the rate of change is not constant over time.
b. The probability of getting into a car accident as a function of the speed at which you drive: This relationship is best described by a non-linear function. As the speed of the vehicle increases, the probability of getting into a car accident initially increases at a slower pace. However, as the speed further increases, the probability of an accident rises more rapidly. This indicates a non-linear relationship since the increase in probability is not constant with respect to speed.
c. The time it takes you to get to work as a function of the speed at which you drive: This relationship can be best described by a linear function. According to the concept of speed = distance/time, as the speed at which you drive increases, the time it takes to get to work decreases linearly. This is because a higher speed leads to covering a greater distance in less time without any other significant factors affecting the relationship.