To solve each problem, refer to the formulas for compound interest. A = P (1 + r/n)^tn and A = Pe^rt Find t, to the nearest hundredth of a year, if \$1786 becomes \$2063 at 2.6%, with interest compounded monthly.

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Identify the given values: the principal amount $P = 1786$, the amount after time $A = 2063$, the annual interest rate $r = 2.6\% = 0.026$, and the compounding frequency $n = 12$ (monthly).

Write down the compound interest formula for interest compounded periodically: $A = P \left(1 + \frac{r}{n}\right)^{nt}$.

Substitute the known values into the formula: $2063 = 1786 \left(1 + \frac{0.026}{12}\right)^{12t}$.

Divide both sides by 1786 to isolate the exponential term: $\frac{2063}{1786} = \left(1 + \frac{0.026}{12}\right)^{12t}$.

Take the natural logarithm of both sides to solve for $t$: $\ln\left(\frac{2063}{1786}\right) = 12t \cdot \ln\left(1 + \frac{0.026}{12}\right)$. Then, solve for $t$ by dividing both sides by $12 \cdot \ln\left(1 + \frac{0.026}{12}\right)$.

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주요 개념

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Compound Interest Formula

The compound interest formula calculates the amount A accumulated over time with principal P, interest rate r, number of compounding periods n per year, and time t in years. It is given by A = P(1 + r/n)^(nt), which accounts for interest being added periodically, causing exponential growth.

Solving for Time in Exponential Equations

To find the time t in compound interest problems, you often need to solve an exponential equation. This involves isolating the exponential term and using logarithms to solve for t, since t appears as an exponent in the formula.

Interest Rate and Compounding Frequency

The interest rate r must be expressed as a decimal, and the compounding frequency n affects how often interest is added. Monthly compounding means n = 12, which influences the growth rate and the calculation of time needed for an investment to reach a certain amount.

To solve each problem, refer to the formulas for compound interest. A = P (1 + r/n)tn and A = Pert Find t, to the nearest hundredth of a year, if \$1786 becomes \$2063 at 2.6%, with interest compounded monthly.