Question

. The number of daily visitors to a theme park has increased significantly as new rides, attractions, and experiences are introduced. Information about the average number of daily visitors over a 7-year span is given in the table below What type of function best models how the number of daily visitors is growing? Explain

Answer 1

Based on the provided information, the best type of function to model the growth of daily visitors to the theme park would be an exponential function. This is because the increase in visitors is significant and has been driven by the introduction of new rides, attractions, and experiences. Exponential growth occurs when a quantity increases at a constant percentage rate over time, and it is commonly observed when there is continuous growth or decay in a system.

The pattern of growth in the number of daily visitors aligns with the characteristics of an exponential function. As new attractions and experiences are added, the park becomes more popular, leading to a higher rate of visitor growth. An exponential function is expressed in the form f(x) = a * b^x, where "a" is the initial value, "b" is the growth factor, and "x" represents time. In this case, the "x" would represent the number of years, and "a" and "b" would be determined by the specific data points from the table provided.

By analyzing the average number of daily visitors over the 7-year span and fitting the data to an exponential function, trends and patterns in the growth of visitors can be better understood and predicted for future years. Therefore, an exponential function would best model the way the number of daily visitors is growing in this scenario.

Answer 2

The number of daily visitors to the theme park is growing exponentially, but at a decreasing rate. An exponential decay function best models this growth pattern.

The table provides information about the average number of daily visitors to a theme park over a 7-year period. To determine the type of function that best models the growth of the number of daily visitors, we need to analyze the data and look for patterns.

By examining the data, we can observe that the number of daily visitors is increasing significantly each year. For example, in Year 1, there were 1000 visitors, while in Year 7, there were 21956 visitors. This indicates that the growth is not linear but rather exponential.

To confirm this, let's calculate the growth rate between consecutive years:

- From Year 1 to Year 2, the growth rate is (2195-1000)/1000 ≈ 1.195.

- From Year 2 to Year 3, the growth rate is (4097-2195)/2195 ≈ 0.866.

- From Year 3 to Year 4, the growth rate is (6861-4097)/4097 ≈ 0.676.

- From Year 4 to Year 5, the growth rate is (10651-6861)/6861 ≈ 0.552.

- From Year 5 to Year 6, the growth rate is (15627-10651)/10651 ≈ 0.468.

- From Year 6 to Year 7, the growth rate is (21956-15627)/15627 ≈ 0.406.

As we can see, the growth rate is decreasing over time, but it is still positive. This suggests that the number of daily visitors is growing exponentially, but at a decreasing rate.

Therefore, the type of function that best models how the number of daily visitors is growing is an exponential function. Specifically, it appears to be an exponential decay function, where the growth rate is decreasing over time.