Problem 81 - Elementary Algebra

Question

Set up an algebraic equation and then solve. A commercial airline pilot flew at an average speed of 350 miles per hour before being informed that his destination airfield may be closed due to poor weather conditions. In an attempt to arrive before the storm, he increased his speed 400 miles per hour and flew for another 3 hours. If the total distance flown was 2,950 miles, then how long did the trip take?

Step-by-Step Solution

Answer

The total trip took 8 hours.

Step 1: Understand the Problem

The pilot has two parts to his trip: one at 350 mph and another at 400 mph. The total distance for the trip is 2,950 miles. We need to find the total time taken for the trip.

Step 2: Define Variables

Let \( t \) represent the time in hours that the pilot flew at 350 mph. Therefore, for the second part at 400 mph, we have already given 3 hours flown.

Step 3: Set Up the Equation for Distance

For the first part of the trip, the distance is \( 350t \) miles. For the second part, the distance is \( 400 \times 3 = 1200 \) miles. The total distance flown is then given by the equation: \( 350t + 1200 = 2950 \).

Step 4: Solve the Equation

Subtract 1200 from each side of the equation to isolate terms with \( t \) on one side:\[350t + 1200 - 1200 = 2950 - 1200\]\[350t = 1750\]Now, divide both sides by 350 to solve for \( t \):\[t = \frac{1750}{350} = 5\]

Step 5: Calculate the Total Trip Time

The time for the first part of the trip is \( t = 5 \) hours. The second part took 3 hours. Therefore, the total trip time is \( 5 + 3 = 8 \) hours.

Key Concepts

Speed Calculation

When dealing with problems related to movement, speed calculation helps us determine how fast an object, like a plane, is moving. Here, speed is typically expressed in units such as miles per hour (mph). The formula to calculate speed is: \[\text{Speed} = \frac{\text{Distance}}{\text{Time}}\] In this exercise, the pilot was flying at constant speeds during different segments of the journey. First, the speed was 350 mph, then increased to 400 mph.Understanding speed in this context requires you to see it not just as numbers, but as the rate at which the plane covered distance over time. This change in speed is crucial because altering speed affects how quickly destinations are reached.

Distance and Time Problem

Distance and time problems are typically solved using relationships between speed, distance, and time. The key here is to understand how these three variables interact.The basic relationship can be described with the formula:\[\text{Distance} = \text{Speed} \times \text{Time}\] In this particular problem, the pilot flew two legs:

• First leg: unknown time \(t\) at 350 mph
• Second leg: 3 hours at 400 mph

The total distance was given as 2,950 miles. By setting up an equation for the total distance using individual parts, we can determine the unknown time spent on the first leg.

Variable Definition

In mathematical problem-solving, defining variables is a fundamental step. It involves choosing symbols to represent unknown quantities, which simplifies the process of forming equations and solving them.For this exercise, we defined the variable \( t \) to represent the time (in hours) the pilot flew at the initial speed of 350 mph. When you define \( t \, \), it sets a foundation for solving the problem algebraically. Instead of approaching each part of the problem independently, using variables indicates what you need to solve for. It helps in setting up equations, like \(350t + 1200 = 2950\), and subsequently finding solutions by manipulating these equations.