Amanda put $1500 in a savings account. After 5 years, she had $1833 in the account. What rate of interest did she earn? Use the formula A = Pert, where A is the ending amount, P is the principal (initial amount), r is the interest rate, and t is time.
A. 5%
B. 3%
C. 20%
D. 4%
Answer (2)
Substitute the given values into the formula A = P e r t .
Isolate the exponential term by dividing both sides by the principal amount.
Apply the natural logarithm to both sides of the equation.
Solve for the interest rate, r , and convert it to a percentage: 4% .
Explanation
Understanding the Problem Let's analyze the problem. We are given the principal amount (P), the ending amount (A), and the time (t). We need to find the interest rate (r) using the formula A = Pet.
Given Information We are given: Principal amount, P = $1500 Ending amount, A = $1833 Time, t = 5 years Formula: A = P e r t , where r is the interest rate.
Substituting Values Substitute the given values into the formula: 1833 = 1500 ⋅ e 5 r
Isolating the Exponential Term Divide both sides by 1500: 1500 1833 = e 5 r
Applying Natural Logarithm Take the natural logarithm of both sides: ln ( 1500 1833 ) = 5 r
Solving for r Divide by 5 to solve for r: r = 5 l n ( 1500 1833 )
Calculating r Calculate the value of r: r = 5 l n ( 1.222 ) ≈ 5 0.2001 ≈ 0.04002
Converting to Percentage Multiply r by 100 to express it as a percentage: r ≈ 0.04002 × 100 = 4.002%
Final Answer The interest rate is approximately 4%. Comparing the calculated percentage with the given options, the closest one is 4%.
Examples
Understanding exponential growth is crucial in finance. For instance, when you invest in a bond, the interest earned follows a similar exponential pattern. If you invest $1000 in a bond with a 6% annual interest rate, you can calculate how much your investment will be worth after a certain period using the same formula, helping you make informed financial decisions. This principle applies to various scenarios like calculating population growth or the decay of radioactive substances.
Amanda earned an interest rate of approximately 4% on her savings account after 5 years, which was calculated using the formula for continuous compound interest. The final answer corresponds to option D. 4%.
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