Problem 56 - Basic Technical Mathematics with Calculus

Question

Solve the given problems. All numbers are accurate to at least two significant digits. An investment of \(\$ 2000\) is deposited at a certain annual interest rate. One year later, \(\$ 3000\) is deposited in another account at the same rate. At the end of the second year, the accounts have a total value of \(\$ 5319.05 .\) The interest rate \(r\) can be found by solving \(2000(1+r)^{2}+3000(1+r)=\$ 5319.05 .\) What is the interest rate?

Step-by-Step Solution

Answer

The interest rate is approximately 0.04 or 4%.

Step 1: Identify the Problem

We are asked to find the interest rate \( r \) such that the total value of two investments matches a given amount at the end of two years.

Step 2: Write the Equation

The equation representing the total value is given by \( 2000(1+r)^2 + 3000(1+r) = 5319.05 \). Our task is to solve for \( r \).

Step 3: Simplify the Equation

Expand the equation to get \( 2000((1+r)^2) + 3000(1+r) = 5319.05 \).First, expand \((1+r)^2 = 1 + 2r + r^2\), so the equation becomes:\[2000(1 + 2r + r^2) + 3000(1 + r) = 5319.05\].

Step 4: Distribute Terms

Distribute the 2000 and 3000 into their respective expressions:\(2000 + 4000r + 2000r^2 + 3000 + 3000r = 5319.05\).

Step 5: Combine Like Terms

Combine all like terms on the left side of the equation:\(2000r^2 + 7000r + 5000 = 5319.05\).

Step 6: Move Constant Terms and Set Equation

Move the constant 5000 to the right side of the equation by subtracting it from both sides:\(2000r^2 + 7000r = 5319.05 - 5000\).This simplifies to:\(2000r^2 + 7000r = 319.05\).

Step 7: Apply the Quadratic Formula

The equation now is quadratically arranged as \(2000r^2 + 7000r - 319.05 = 0\). Use the quadratic formula where \(a = 2000\), \(b = 7000\), \(c = -319.05\):\[r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\].Calculate each part to find \(r\).

Step 8: Calculate Discriminant

Find the discriminant: \(b^2 - 4ac = 7000^2 - 4(2000)(-319.05)\).Calculate this to continue solving.

Step 9: Find Possible Values for r

Compute the discriminant and use it in the formula to find possible values for \(r\).Complete the formula calculations to find the correct value for \(r\).

Step 10: Select and Verify Solution

Choose the realistic value for \(r\) (typically the positive root), and check the answer by substituting back into the original problem to ensure it solves the equation.

Key Concepts

Interest Rate Calculation

Interest rate calculation might seem daunting at first, but it's quite systematic. Interest is essentially the cost of borrowing money or the reward for depositing money, expressed as a percentage over time. When calculating interest, you often encounter the following concepts:* **Principal**: The original sum of money placed into an account or borrowed.* **Rate (r)**: The percentage of the principal charged as interest each period.* **Time**: How long you'll be calculating the interest for.Most often, when calculating for investments or loans, you will use either simple or compound interest. In our exercise, we're dealing with compound interest, which accounts for interest on both the initial principal and the accumulative interest from previous periods:\[ V = P(1 + r)^n \]This formula represents the future value \( V \) of the investment or loan, where \( P \) is the principal, \( r \) is the interest rate expressed as a decimal, and \( n \) is the number of compounding periods.By rearranging these variables and solving the equation, you can find the missing piece, which is often the interest rate \( r \) in such problems.

Solving Equations

Solving equations is all about finding the value or values of unknown variables that make the equation true. Equations represent a balance; both sides should equal each other. In our exercise, we aimed to find the interest rate \( r \) that balances the given equation:\[ 2000(1+r)^2 + 3000(1+r) = 5319.05 \]Here’s how you typically approach solving an equation:1. **Simplify each side**: Begin by expanding and simplifying each side of the equation as much as possible. Combine like terms.2. **Isolate the variable**: Rearrange the equation to get the unknown variable (in this case \( r \)) by itself on one side of the equation.For our quadratic equation, this involves expanding expressions, distributing terms, and moving constants. The goal is to end up with an equation in the standard quadratic form:\[ ax^2 + bx + c = 0 \]3. **Solve for the variable**: Use appropriate methods to solve for the variable. For standard quadratic equations, you can use factoring, completing the square, or the quadratic formula.4. **Verify your solution**: Substitute your solution back into the original equation to ensure both sides are equal.In this way, solving complex equations becomes a step-by-step process that ensures accuracy and learning.

Discriminant

The discriminant is a key element of the quadratic equation formula, used to determine the nature of the roots without actually solving the equation. It is part of the quadratic formula:\[ r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]The discriminant itself is given by the expression \( b^2 - 4ac \). This value provides crucial information about the roots:* If the discriminant is **positive**, there are two distinct real roots.* If it is **zero**, there is exactly one real root, meaning the quadratic touches the x-axis at one point.* If it is **negative**, there are no real roots, but rather two complex roots.In our example, using the discriminant helps predict whether the interest rate will have one viable real-world solution (the positive, realistic root) or two. Calculating the discriminant gives an idea of what to expect without exhaustive computation. This powerful tool facilitates decision-making in financial contexts by indicating possible outcomes for investments and loans.