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. (4 points)
notice for Option 1: 1100 1210 1331
1210 / 1100 = 1331 / 1210 = 1.1
so we have a geometric progression, making the
function exponentinal
term(n) = 1100 (1.1)^(n-1) , where n is the number of years
in Option 2: 1100 1200 1300
we have 1200 - 1100 = 1300 - 1200 = 100
we have a common difference which gives us an arithmetic progression or a linear function
term(n) = 1100 + 100(n-1)
= 1000 + 100n , where n is the number of years
so you want sum(20) for each case
for option 1:
sum(20) = 1100( 1.1^20 - 1)/ .1 = 63002.50
for option 2:
sum(20) = (20/2)(2(1100) + 19(100)) = 41000.00
Whoa
Part A: To determine whether the function is linear or exponential, we need to examine the rate at which the value of the investment changes over time.
For Option 1, we observe that the value increases by $110 each year. This increase remains constant regardless of the number of years.
On the other hand, for Option 2, the value increases by $100 in the first year, $90 in the second year, and $100 in the third year. The rate of increase is not constant, and it decreases over time.
Based on this analysis, we can conclude that Option 1 follows a linear function, while Option 2 follows a decreasing exponential function.
Part B: For Option 1, since the value increases by $110 each year, we can represent the value after n years using the linear function:
f(n) = 1000 + 110n
For Option 2, we can observe that the value decreases by a certain percent each year. To determine this percent, we can calculate the difference in value from the previous year to the current year:
Year 2: $1200 - $1100 = $100 increase
Year 3: $1300 - $1200 = $100 increase
We can see that the value increases by $100 each year, which implies a constant annual percent increase of 10%. Therefore, we can represent the value after n years for Option 2 using the exponential function:
f(n) = 1000 * (1 + 10/100)^n
= 1000 * (1.1)^n
Part C: To determine which option would help Belinda increase her investment value by the greatest amount in 20 years, we can calculate the value for each option after 20 years.
For Option 1:
f(20) = 1000 + 110 * 20
= 1000 + 2200
= $3200
For Option 2:
f(20) = 1000 * (1.1)^20
≈ 1000 * 6.7275
≈ $6728
By comparing the values, we can see that Option 2 yields a significantly higher investment value after 20 years compared to Option 1. The difference is approximately $3528. Therefore, if Belinda wants to maximize her investment value after 20 years, she should choose Option 2.