Problem 58 - Algebra A Combined Function

Question

How long will it take a bus traveling at 60 miles per hour to overtake a car traveling at \(40 \mathrm{mph}\) if the car had a 1.5 -hour head start?

Step-by-Step Solution

Answer

The bus will overtake the car in 3 hours.

Step 1: Understand the Problem

The bus and the car are traveling at different speeds, and the car had a 1.5-hour head start. Find out when the distance traveled by the bus equals the distance traveled by the car.

Step 2: Define Variables

Let the time the bus travels be \(t\) hours. The car has already traveled for 1.5 hours before the bus starts.

Step 3: Calculate Distance Traveled by the Car

The car moves at a speed of 40 mph and travels for \(t + 1.5\) hours. Thus, the distance it travels is given by \(40(t + 1.5)\).

Step 4: Calculate Distance Traveled by the Bus

The bus travels at 60 mph for \(t\) hours. So, the distance the bus travels is \(60t\).

Step 5: Set Up the Equation

For the bus to overtake the car, the distance covered by both should be the same. Set up an equation: \(60t = 40(t + 1.5)\).

Step 6: Solve the Equation

Distribute and simplify the equation: \(60t = 40t + 60\). Subtract \(40t\) from both sides: \(20t = 60\). Divide both sides by 20: \(t = 3\).

Step 7: Conclusion

The bus will take 3 hours to overtake the car.

Key Concepts

Algebraic Equations

Algebraic equations are essential tools for solving problems involving distances and speeds. They help us express relationships and uncover unknown values using mathematical symbols and operations. In the context of a word problem, an algebraic equation is used to express the equality that marks an important condition or result. Here, our goal is to determine the time it takes for a bus traveling at 60 miles per hour to catch up with a car moving at 40 miles per hour, which had a head start.

An algebraic equation forms when we set two expressions equal to each other. In this problem, the equation is established when the distances traveled by the bus and the car are equal. We represent this condition as an equation, utilizing known speeds and the unknown time variable. It captures a balance or equality needed to find the solution to the given scenario. For example, the equation from our step-by-step solution above is:

• Given equation: \[ 60t = 40(t + 1.5) \]

This equation sets the distances equal, forming the basis for finding the unknown time \(t\). Solving such equations often involves operations like distribution, addition, subtraction, multiplication, or division, which are necessary to isolate the unknown variable.

Variables and Expressions

Variables and expressions are fundamental components that we use extensively in algebra to simplify and solve problems. In the given exercise, a variable represents an unknown quantity that needs to be determined. It acts as a placeholder in algebraic expressions and equations.

In this exercise, the variable \(t\) represents the time in hours that the bus travels. We use expressions to calculate the distances based on given speeds and time periods. An expression combines numbers, variables, and operations into a meaningful representation. In our distance problem, these expressions represent the distance each vehicle travels:

• Distance by car: \[ 40(t + 1.5) \]
• Distance by bus: \[ 60t \]

The expression \(40(t + 1.5)\) takes into account the car's head start by adding 1.5 to the bus's travel time. This demonstrates the importance of expressions in encapsulating real-world scenarios mathematically.

Problem Solving Steps

Approaching a problem methodically makes solving it much simpler and more structured. Breaking down a complex scenario into coherent steps helps identify the necessary information, determine relationships, and find a solution.

**Understanding the Problem:** The first step is to fully comprehend what the problem asks. In this exercise, we identify the bus and the car's speeds, the head start, and the need to equalize the travel distances to find when the bus overtakes the car.

• Identify all given information and what needs to be found.

**Define Variables and Equations:** The second step is to assign variables for unknowns and set up equations. For the bus and car, we define the time \(t\) and express distances in terms of \(t\). The equations stem from understanding the relationship between travel time and speed.

• Set up equations using defined variables.

**Solve and Conclude:** Finally, solve the equations using algebraic techniques, ensuring each step maintains balance between both sides. The solution gives a clear answer to the question. Here, we found that the bus takes 3 hours to overtake the car.

• Perform necessary mathematical operations to solve for the unknown.
• Draw your final conclusion from the solved equations.

These structured steps not only streamline the problem-solving process but also make it easier to tackle similar problems in the future.