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)
option 1 is not linear because it does not change by the same amount every year. It does not have a constant slope.
option 2 increases by 100 per year, constant slope, straight line, linear
-----------------------
a = A y^n
1100 =A y
1210 = Ay^2
1331 = A y^3
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1210/1100 = y = 1.1
check 1331/1210 = y = 1.1 sure enough
1100 = A (1.1)
A = 1000
so
a = 1000 * 1.1^n
and the linear option 2
a = 1000+100n
You can put the numbers in for part C
10% compounded is better than 10% simple :)
no
was this right?
I need help
Part A: Belinda's investment value under option 1 increases by a fixed amount each year (100) and follows a linear pattern. On the other hand, her investment value under option 2 follows an exponential pattern, as it increases by a fixed percentage each year (10%). Therefore, the function to describe the value of the investment after a fixed number of years using option 1 is linear, while the function for option 2 is exponential.
Part B: The function for option 1 can be written as f(n) = 100n + 1000, where n represents the number of years.
The function for option 2 can be written as f(n) = 100(1.1)^n + 1000, where n represents the number of years.
Part C: In 20 years, the value of Belinda's investment under option 1 would be f(20) = 100(20) + 1000 = 3000 dollars.
Under option 2, the value of Belinda's investment after 20 years would be f(20) = 100(1.1)^20 + 1000 ≈ 6727.51 dollars.
Therefore, there would be a significant difference in the value of Belinda's investment after 20 years if she chooses option 2 over option 1. The value under option 2 (approximately $6727.51) would be considerably higher than under option 1 (only $3000).
Part A: To determine whether a linear or exponential function can be used to describe the value of the investment, we need to examine how the values change over time. A linear function describes a constant rate of change, while an exponential function describes a growth rate that increases or decreases exponentially.
Looking at the table, we can see that the value of the investment under option 1 increases by an additional $110 per year, while the value under option 2 increases by an additional $100 per year. Since the amount of increase is constant for both options, we can conclude that the value of the investment under option 1 follows a linear function, while the value under option 2 also follows a linear function.
Part B:
For option 1, the value of the investment can be described by the function: f(n) = 1000 + 110n, where n represents the number of years.
For option 2, the value of the investment can be described by the function: f(n) = 1000 + 100n, where n represents the number of years.
Part C: To determine which option would result in a greater investment value after 20 years, we can calculate the value using the given functions.
For option 1: f(20) = 1000 + 110 * 20 = 1000 + 2200 = 3200 dollars.
For option 2: f(20) = 1000 + 100 * 20 = 1000 + 2000 = 3000 dollars.
From the calculations, we can see that the value of Belinda's investment after 20 years would be $3200 under option 1 and $3000 under option 2. Therefore, there would be a significant difference of $200 in the value of Belinda's investment after 20 years if she chooses option 1 over option 2.