Problem 20 - College Algebra

Question

For exercises 17 and 18, use this scenario: A retired woman has \(\$ 50,000\) to invest but needs to make \(\$ 6,000\) a year from the interest to meet certain living expenses. One bond investment pays 15\(\%\) annual interest. The rest of it she wants to put in a CD that pays 7\(\% .\) Ben starts walking along a path at 4 \(\mathrm{mi} / \mathrm{h}\) . One and a half hours after Ben leaves, his sister Amanda begins jogging along the same path at 6 \(\mathrm{mi} / \mathrm{h}\) . How long will it be before Amanda catches up to Ben?

Step-by-Step Solution

Answer

Amanda catches up to Ben in 3 hours after she starts.

Step 1: Define the scenario and relations

We need to find out when Amanda catches up to Ben. Initially, Ben starts walking at 4 miles per hour. After 1.5 hours, Amanda starts jogging at 6 miles per hour on the same path. When Amanda catches Ben, they will have traveled the same distance from their respective starting points.

Step 2: Determine Ben's head start distance

Since Ben has a head start of 1.5 hours, we calculate the distance he covers during this time. The distance Ben covers can be found by multiplying his speed by time, which gives us: \( 4 \text{ mi/h} \times 1.5 \text{ h} = 6 \text{ miles} \). So Ben is 6 miles ahead when Amanda starts.

Step 3: Set up equations for their distances

Let \( t \) be the time (in hours) it takes for Amanda to catch up after she starts jogging. In \( t \) hours, Amanda will cover \( 6t \) miles since she travels at 6 \( \mathrm{mi/h} \). Ben will have traveled an additional \( 4t \) miles in this time, making his total distance from the start \( 6 + 4t \) miles because he had a 6-mile head start.

Step 4: Apply the concept of equal distances

For Amanda to catch up with Ben, she must have traveled the same total distance as Ben. Therefore, set Amanda's distance equal to Ben's distance: \( 6 + 4t = 6t \).

Step 5: Solve the equation for t

Simplify and solve the equation: \( 6 + 4t = 6t \). Moving the terms around gives: \( 6 = 6t - 4t \), simplifying further gives: \( 6 = 2t \). Divide both sides by 2 to find \( t \): \( t = 3 \). Thus, it will take 3 hours for Amanda to catch up to Ben once she starts jogging.

Key Concepts

Investment Scenarios

Investment scenarios often involve deciding how to allocate funds to achieve a certain financial outcome. In the case of the retired woman, she has $50,000 to invest and needs $6,000 annually from the interest to meet her living expenses. This means she must consider various investment options with different interest rates to reach her goal.

Here's how investing works:

• Each investment option has an interest rate, expressed as a percentage of the principal amount.
• The total return from all investments should add up to at least the desired annual income.
• The woman chooses a bond paying 15% interest and a CD offering 7% interest.

To determine how much should be allocated to each investment, use the linear equation method. This method helps balance the varying interest rates to meet the financial target. It's a practical approach to handling such investment decisions effectively.

Time and Distance Problems

Time and distance problems are common in algebra and involve calculating how long it takes for objects moving at different speeds to meet or compare distances. In the original exercise, Ben and Amanda are tackling such a problem.

Essential aspects of these problems include:

• The speed at which each person or object travels.
• The time each starts moving, including any head start involved.
• The distance covered by each person or object to determine when an intersection occurs.

In this scenario, Ben leaves first, and Amanda starts later but travels faster. Calculating the point at which Amanda catches up requires setting the distance they travel equal. This often leads to forming and solving linear equations, making this type of problem a practical application of algebra.

Linear Equations

Linear equations are fundamental in solving problems involving unknown values and relationships between different quantities. They are expressed in the form \( ax + b = 0 \), where \( x \) is the variable to be solved.

In the original problem, a linear equation is used to determine when Amanda will catch up with Ben:

• First, equate the distance traveled by both individuals since catching up implies equal distances.
• Create an equation based on their respective speeds and the time or head start involved.
• Solve the equation by simplifying and isolating the variable to find the time or distance value.

Understanding how to set up and solve linear equations helps tackle various real-world problems, from investment scenarios to time and distance challenges. They serve as a bridge connecting situational details to practical solutions.