Jenny invests $2,000 at an interest rate of 5%. The amount of money, mc007-1.jpg, in Jenny’s account after t years can be represented using the equation mc007-2.jpg. If Jenny would have invested the same amount of money at the same interest rate four years ago, the equation representing the amount of money, mc007-3.jpg, in her account would be represented using the equation mc007-4.jpg. Which of the following is equivalent to mc007-5.jpg?
To find an equivalent expression for mc007-5.jpg, we first need to understand the given information.
According to the problem, Jenny invested $2,000 at an interest rate of 5%. The amount of money in her account after t years can be represented by the equation mc007-2.jpg.
The equation mc007-2.jpg represents the future value (FV) of her investment, calculated using the compound interest formula:
FV = P(1 + r/n)^(n*t)
Where:
FV is the future value
P is the principal amount (initial investment)
r is the annual interest rate (expressed as a decimal)
n is the number of times interest is compounded per year
t is the number of years
In this case, we have P = $2,000, r = 0.05 (5%), and n = 1 (compounded annually). So, the equation mc007-2.jpg becomes:
FV = 2000(1 + 0.05)^t
Now, let's consider the scenario where Jenny had invested the same amount four years ago. To find the equation representing the amount of money in her account (mc007-3.jpg), we need to adjust the time (t).
Since Jenny invested four years ago, the new value for t will be t - 4. Therefore, the equation becomes:
FV = 2000(1 + 0.05)^(t - 4)
Now, we want to find an equivalent expression for mc007-5.jpg. Looking at the equation mc007-3.jpg, we can see that the expression inside the parentheses is (1 + 0.05)^(t - 4). To simplify it further, we can expand the exponent using the properties of exponents:
(1 + 0.05)^(t - 4) = (1.05)^(t - 4)
Hence, mc007-5.jpg is equivalent to (1.05)^(t - 4).
To find an equivalence to mc007-5.jpg, we need to first determine the equation representing the amount of money in Jenny's account after 4 years.
Using the formula for compound interest, the equation representing the amount of money in Jenny's account after t years is given by:
A = P(1 + r)^t
Where:
A = the amount of money in the account after t years
P = the initial investment
r = the interest rate (expressed as a decimal)
t = the number of years
Now, let's calculate the amount of money in Jenny's account after 4 years using this equation:
A = 2000(1 + 0.05)^4
A = 2000(1.05)^4
A = 2000(1.21550625)
A ≈ 2431.01
Therefore, the equation representing the amount of money in Jenny's account after 4 years is:
mc007-3.jpg = 2431.01
Now, we can compare this with the given equation mc007-4.jpg:
mc007-4.jpg = (2000(1 + 0.05))^(-4)
To find an equivalent equation to mc007-5.jpg, we need to simplify mc007-4.jpg. We can do this by applying the exponent rule, which states that (a^b)^c = a^(b*c):
mc007-4.jpg = (2000(1.05))^(-4)
mc007-4.jpg = 2000^(1*(-4))(1.05)^(-4)
mc007-4.jpg = 2000^(-4)(1.05)^(-4)
Now, we can see that mc007-4.jpg is equivalent to:
mc007-4.jpg = mc007-5.jpg
Therefore, the answer is mc007-5.jpg.