Answer: it will take approximately 14.62 years for the initial investment to double at an interest rate of 4.75% per year, compounded continuously.
Step-by-step explanation: Given an investment of $2000 at a continuously compounded interest rate of 4.75%, the balance in the account can be calculated using the following mathematical expression after t years:
The aforementioned equation, A = P * e^(rt), denotes the relationship between the accrued amount (A) and the principal amount (P), compounded continuously at a fixed annual rate of interest (r) over a specific time period (t), as governed by the mathematical constant "e."
In the context of financial calculations, the symbol 'P' denotes the initial capital investment. The interest rate, represented by the variable 'r', is expressed in the form of a decimal. Additionally, 'e' is the mathematical constant, roughly equivalent to 2.71828. Finally, 't' refers to the duration of the investment, measured in years.
In order to determine the duration of time required for the investment to achieve a twofold increase, it is necessary to solve the corresponding equation:
The equation expressed as 2P = P * e^(rt) can be restated more formally as follows. Given a principal investment amount represented by P and a rate of return indicated by r, compounded over time t, the equation can be expressed as the product of P and the exponential function of e^(rt), yielding twice the initial investment amount.
The variable 2P represents the monetary value acquired through doubling the initial investment.
Upon division of both sides by P, the resulting expression is as follows:
The equation 2 equals the exponential function of the base e raised to the power of the product of r and t.
By applying the natural logarithm function to both expressions, the resultant outcome is:
The natural logarithm of 2 can be represented as rt, where r denotes the logarithmic base and t denotes the logarithm of the argument, in accordance with the conventions of academic mathematical writing.
Upon resolving for the variable t, an outcome is yielded:
The mathematical expression t = ln(2) / r can be written in a formal academic style as follows: The equation determines the relationship between time t and the rate of decay r, where t is equal to the natural logarithm of 2 divided by r.
Upon substitution of the provided values, the resultant output is:
The calculated value of the variable t, representing the length of time in years, is approximately equal to 14.62 years, obtained through the algebraic manipulation of the natural logarithmic function of 2 divided by the constant value of 0.0475.