8. 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 thoroughly

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 (from age 0 to 100):

In this relationship, it is best described by a non-linear function. Typically, a person's height increases as they age, but not at a constant rate. We observe rapid growth in the early years of life, then the growth gradually slows down and eventually stops. This suggests that the relationship between height and age is not linear, as a linear function implies a constant rate of change. Instead, we can expect a curve-like shape in the relationship. Specifically, the growth rate is higher during childhood but gradually decreases as the person reaches adulthood until it stops altogether.

To further analyze this relationship, we can gather data on individuals' heights at different ages, plot them on a graph, and observe the trend. If the plotted points form a curve rather than a straight line, it would provide strong evidence that the relationship is non-linear.

b. The probability of getting into a car accident as a function of the speed at which you drive:
This relationship can be best described by a non-linear function. When driving at lower speeds, the probability of getting into a car accident is generally lower. As the speed increases, the risk of accidents also tends to increase, but not in a proportionate manner. At extremely high speeds, the risk increases significantly, indicating a non-linear relationship.

We can gather data on accident probabilities at different driving speeds and plot them on a graph. If the data points exhibit a non-linear pattern, such as accelerating growth as speed increases, it would support the argument for a non-linear relationship.

c. The time it takes you to get to work as a function of the speed at which you drive:
This particular relationship can be best described by a linear function. The time it takes to get to work is directly influenced by the speed at which you drive. As you increase your speed, the time it takes to reach your destination will decrease proportionally. For example, if you double your speed, you will reach your destination in half the time.

To confirm this relationship, we can conduct an experiment by driving at different speeds and measuring the corresponding time it takes to reach work. By plotting these data points on a graph, we should see a straight line with a constant slope. This linear pattern would indicate a linear relationship between the time taken and the driving speed.