Problem 53 - Beginning and Intermediate Algebra

Question

Solve each problem using a quadratic equation. The formula \(A=P(1+r)^{2}\) gives the amount \(A\) in dollars that \(P\) dollars will grow to in 2 yr at interest rate \(r\) (where \(r\) is given as a decimal), using compound interest. What interest rate will cause \(\$ 2000\) to grow to \(\$ 2142.45\) in \(2 \mathrm{yr} ?\)

Step-by-Step Solution

Answer

The interest rate is 3.5%.

Step 1: Identify the given values

Given: - Principal amount, \( P = 2000 \)- Amount after 2 years, \( A = 2142.45 \)

Step 2: Set up the equation with given values

The formula is \( A = P(1+r)^2 \). Substitute the known values:\[ 2142.45 = 2000(1+r)^2 \]

Step 3: Isolate the \( (1+r)^2 \) term

Divide both sides of the equation by 2000:\[ \frac{2142.45}{2000} = (1+r)^2 \]This simplifies to:\[ 1.071225 = (1+r)^2 \]

Step 4: Solve for \( 1+r \)

Take the square root of both sides to solve for \( 1+r \):\[ \sqrt{1.071225} = 1+r \]\[ 1.035 = 1+r \]

Step 5: Solve for \( r \)

Subtract 1 from both sides to isolate \(r\):\[ r = 1.035 - 1 \]\[ r = 0.035 \]

Step 6: Convert \(r\) to a percentage

Since \(r\) is given as a decimal, convert \(0.035\) to a percentage: \[ r = 0.035 \times 100 \% = 3.5\% \]

Key Concepts

quadratic equation

A quadratic equation is a type of polynomial equation of the form \[ ax^2 + bx + c = 0, \] where * \( a, b, \) and \( c \) are constants * \( x \) is the variable. In solving real-world problems, a quadratic equation can emerge naturally when dealing with areas, projectile motion, and, as in our exercise, compound interest calculations.To solve a quadratic equation, various methods like factoring, completing the square, or using the quadratic formula \[ x = \frac{-b \,\pm\, \sqrt{b^2 - 4ac}}{2a} \] can be employed. Breaking down the given exercise: the equation \[ 2142.45 = 2000(1+r)^2 \] resembles a quadratic equation which, upon isolating and taking the square root, simplifies our task to finding \( r \). Large equations can often seem daunting, but by breaking them down step by step, they become manageable.

compound interest

Compound interest is the calculation of interest on the initial principal, which also includes future interest added on that principal.The formula for compound interest is \[ A = P(1 + r)^n, \] where: * \( A \) is the amount of money accumulated after n periods, including interest.* \( P \) is the principal amount (initial amount of money).* \( r \) is the annual interest rate (expressed as a decimal).* \( n \) is the number of times the interest is compounded per year.In our exercise, the formula simplifies to \[ A = P(1+r)^2 \] because we are dealing with an interest compounded annually and a period of 2 years.Starting with the known values, we substitute into the formula and solve for \( r \). This demonstrates how compound interest grows not just on the initial principal but also on the accumulated interest from previous periods.

interest rate calculation

Calculating the interest rate is a common task in finance that helps in forecasting future finances. The interest rate (\( r \)) represents the percentage at which the principal (P) grows over time in an investment or loan scenario. Knowing how to isolate and solve for \( r \) equips you with the ability to determine worth or cost over time. In our exercise, we used the equation \[ 2142.45 = 2000(1+r)^2 \], rearranged it, and isolated \( r \) by solving step-by-step: 1. Divide both sides by the principal (2000), which simplifies our equation: \( \frac{2142.45}{2000} = (1+r)^2 \rightarrow 1.071225 = (1+r)^2 \). 2. Taking the square root of both sides: \( \rightarrow 1.035 = 1 + r \). 3. Subtracting 1 from both sides: \( r = 0.035 \). Lastly, convert this into a percentage by multiplying it by 100: \( r = 0.035 \times 100 \rightarrow 3.5\text{\textpercent} \). This gives us the annual interest rate necessary to grow an investment of \(2000 to \)2142.45 in two years.