nolan began a savings account three years ago. he invested $100 at a 2% interest rate according to the…

nolan began a savings account three years ago. he invested $100 at a 2% interest rate according to the equation $v_n = 100(1.02)^x$, where $v_n$ is the value of his account after $x$ years. anias started an account today. she invested $100 at a 2% interest rate according to the equation $v_a = 100(1.02)^{x - 3}$, where $v_a$ is the value of her account. lets say anias started saving at the same time nolan did, three years ago. approximately how much money would she have had to invest to have the same amount of money she has now?\n$5.77\n$90.24\n$94.23\n$106.12

nolan began a savings account three years ago. he invested $100 at a 2% interest rate according to the equation $v_n = 100(1.02)^x$, where $v_n$ is the value of his account after $x$ years. anias started an account today. she invested $100 at a 2% interest rate according to the equation $v_a = 100(1.02)^{x - 3}$, where $v_a$ is the value of her account. lets say anias started saving at the same time nolan did, three years ago. approximately how much money would she have had to invest to have the same amount of money she has now?\n$5.77\n$90.24\n$94.23\n$106.12

Answer

Explanation:

Step1: Let's assume the current year is (x). Nolan's account value is (V_n = 100(1.02)^x). Anias' account value is (V_a=100(1.02)^{x - 3}).

If Anias started at the same time as Nolan ((x) years ago), let the initial - investment be (P). Then her account value after (x) years would be (V_a = P(1.02)^x). Since (V_a = 100(1.02)^{x - 3}) and (V_a = P(1.02)^x), we can set up the equation: [P(1.02)^x=100(1.02)^{x - 3}]

Step2: Divide both sides of the equation by ((1.02)^x).

We get (P = 100\times(1.02)^{x - 3}\div(1.02)^x). Using the exponent rule (a^m\div a^n=a^{m - n}), we have (P = 100\times(1.02)^{(x - 3)-x}=100\times(1.02)^{- 3}).

Step3: Calculate ((1.02)^{-3}).

((1.02)^{-3}=\frac{1}{(1.02)^3}). And ((1.02)^3=1.02\times1.02\times1.02 = 1.061208). So (\frac{1}{(1.02)^3}=\frac{1}{1.061208}\approx0.9423).

Step4: Calculate (P).

(P = 100\times0.9423 = 94.23).

Answer:

($94.23)