3.Belinda wants to invest $1000. The table below shows the value of her investment under two different options for three different years:
Number of years
1 2 3
Option 1 (amount in dollars)
1100 1210 1331
Option 2 (amount in dollars) 1100 1200 1300
Part A: What type of function, linear or exponential, can be used to describe the value of the investment after a fixed number of years using option 1 and option 2? Explain your answer. (2 points)
Part B: Write one function for each option to describe the value of the investment f(n), in dollars, after n years. (4 points)
Part C: Belinda wants to invest in an option that would help to increase her investment value by the greatest amount in 20 years. Will there be any significant difference in the value of Belinda's investment after 20 years if she uses option 2 over option 1? Explain your answer, and show the investment value after 20 years for each option.
Part A: To determine what type of function can be used to describe the value of the investment under each option, we need to examine the data given. In option 1, the values increase by a fixed amount each year (i.e., 1100, 1210, 1331), while in option 2, the values increase by a constant percentage each year (i.e., 1100, 1200, 1300).
Based on this observation, we can conclude that option 1 follows a linear function since the amount increases by a constant amount each year. By contrast, option 2 follows an exponential function since the amount increases by a constant percentage each year.
Part B: To write the functions for each option, we need to determine the relationship between the number of years (n) and the value of the investment (f(n)).
For option 1, since the value increases by a fixed amount of $1100 each year, we can use the linear equation:
f(n) = 1000 + 1100n
Here, the initial investment is $1000, and n represents the number of years.
For option 2, the value increases by a constant percentage each year. As the percentage increase is not explicitly stated, we can determine it by finding the common ratio between consecutive values. From the given data, we can see that the values increase by 100 each year (i.e., 1100 to 1200, and 1200 to 1300). Thus, the common ratio is 1.1 (i.e., 1100 * 1.1 = 1210, 1210 * 1.1 = 1331).
Therefore, the exponential function for option 2 is:
f(n) = 1000 * (1.1)^n
Part C: To determine if there will be a significant difference in the value of Belinda's investment after 20 years between option 1 and option 2, we can substitute n = 20 into the respective functions and compare the results.
For option 1:
f(20) = 1000 + 1100 * 20
= 1000 + 22000
= 23000
For option 2:
f(20) = 1000 * (1.1)^20
≈ 6727.50
By comparing these values, it is evident that there will be a significant difference in the value of Belinda's investment after 20 years if she chooses option 2 instead of option 1. Option 1 results in an investment value of $23,000, while option 2 provides an investment value of approximately $6,727.50.