Problem 42 - Precalculus with Limits

Question

Airplane Speed Two planes start from Los Angeles International Airport and fly in opposite directions. The second plane starts \(\frac{1}{2}\) hour after the first plane, but its speed is 80 kilometers per hour faster. Find the airspeed of each plane when 2 hours after the first plane departs the planes are 3200 kilometers apart.

Step-by-Step Solution

Answer

The airspeed of the first plane is 880 km/h and the airspeed of the second plane is 960 km/h.

Step 1: Define the variables

Let's denote the airspeed of the first plane as \(x\) km/h. Since the second plane is moving at a speed that is 80km/h faster, its speed is \(x + 80\) km/h.

Step 2: Apply the distance formula

The first plane has been flying for 2 hours, so the distance it covers is \(2x\) km. The second plane has been flying for 1.5 hours (since it started half an hour later), so its distance is \(1.5(x+80)\) km.

Step 3: Set up the equation

The total distance covered by both planes is given as 3200km. Therefore, we set up the equation as: \(2x + 1.5(x+80) = 3200\)

Step 4: Solve the equation

Expanding and simplifying the equation get us: \(2x + 1.5x + 120 = 3200\), then combining like terms we get \(3.5x = 3080\). Dividing both sides by 3.5, we get \(x = 880\) km/h.

Step 5: Find the speed of the second plane

From step 1, we know that the speed of the second plane is \(x + 80\). So substituting \(x = 880\) km/h, we get the speed of the second plane as \(880 + 80 = 960\) km/h.

Key Concepts

Distance Formula

The distance formula is the tool we use to calculate how far an object has moved, based on its speed and the time it has been traveling. In this exercise, it helps us determine how far each plane has flown since they started from Los Angeles International Airport.
To use the distance formula, remember the simple equation:

• Distance = Rate × Time

In our airplane problem, this formula varies slightly for each plane due to different starting times. The first plane flies for the entire 2 hours, while the second flies for 1.5 hours (since it started 30 minutes later). Their rates, or speeds, are defined as variables based on the given conditions. By plugging their speeds and times into the formula, we can calculate the distance each plane covers before setting up an additional equation to find each plane's airspeed.

Rate and Time

Rate and time are crucial components when dealing with distance problems. In this exercise, the two planes travel at different rates due to differing airspeeds and start times.

• Rate (or speed) is how fast an object moves, usually given in kilometers per hour (km/h) in this context.
• Time is how long the object moves, typically measured in hours.

Understanding how these two elements interact is key to solving the problem. The first plane's rate is defined as a variable, while the second plane's rate is the first plane’s speed plus an additional 80 km/h.
For time, since the first plane has a half-hour head start, the time for each is slightly different. These differences help us create the distances they travel, which ultimately provide the solution with more data points to work from.

Solving Equations

Solving equations can be straightforward when all the pieces are correctly accounted for. Here, setting up the equation involves using the sum of distances traveled by each plane, equating it to the total distance apart given (3200 km).
To solve such an equation:

• Begin with setting up an equation that reflects the actual scenario. In our case: \[2x + 1.5(x + 80) = 3200\]
• Combine like terms; simplify the expression: \[2x + 1.5x + 120 = 3200\]
• Then, solve for the unknown variable by isolating it on one side of the equation, resulting in: \[3.5x = 3080\]
• Finally, divide to find the individual rate of the first plane: \[x = \frac{3080}{3.5} = 880\] km/h

By following these steps closely, we can discover the airspeed of each plane.