Question

Belinda wants to invest $1,000. 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) 1300 1690 2197 Option 2 (amount in dollars) 1300 1600 1900 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. (4 points)

Answer 1

Explanation:

Part A:

Option 1 shows an exponential growth pattern, as the value of the investment is increasing at an increasing rate each year. Option 2, on the other hand, shows a linear growth pattern, as the value of the investment is increasing at a constant rate each year.

Part B:

Option 1 can be described by the exponential function:

f(n) = 1000 x (1.3)^n

where n is the number of years, and 1.3 is the growth factor (calculated as 1300/1000).

Option 2 can be described by the linear function:

f(n) = 1000 + 300n

where n is the number of years, and 300 is the constant rate of increase in dollars per year.

Part C:

To determine which option will increase Belinda's investment value by the greatest amount in 20 years, we can calculate the value of each option after 20 years using the respective functions:

For Option 1:

f(20) = 1000 x (1.3)^20

f(20) = 1000 x 6.1917

f(20) = 6,191.70

For Option 2:

f(20) = 1000 + 300(20)

f(20) = 1000 + 6000

f(20) = 7,000

So, the investment value for Option 2 after 20 years is higher than that of Option 1. The difference in value is significant, as Option 2 would give Belinda an investment value of $7,000, while Option 1 would give her an investment value of $6,191.70. Therefore, if Belinda wants to increase her investment value by the greatest amount in 20 years, she should choose Option 2.