Problem 13 - Algebra 2

Question

Flight instruction costs \(\$ 105\) per hour, and the simulator costs \(\$ 45\) per hour. Hai-Ling spent 4 more hours in airplane training than in the simulator. If Hai-Ling spent \(\$ 370\) , how much time did he spend training in an airplane and in a simulator?

Step-by-Step Solution

Answer

Hai-Ling did not record the costs or hours correctly; re-evaluate initial conditions.

Step 1: Define Variables

Let \( x \) be the number of hours spent in the simulator, and since Hai-Ling spent 4 more hours in airplane training, let \( x + 4 \) be the number of hours spent in airplane training.

Step 2: Set Up Equation for Total Cost

The cost for simulator training is \( 45x \) and for airplane training is \( 105(x + 4) \). The total cost is given as \( 370 \). We can set up the equation: \[ 45x + 105(x + 4) = 370 \]

Step 3: Simplify and Solve the Equation

Expand and simplify the equation: \[ 45x + 105x + 420 = 370 \]Combine like terms: \[ 150x + 420 = 370 \]Subtract 420 from both sides: \[ 150x = -50 \]Divide both sides by 150 to solve for \( x \): \[ x = \frac{-50}{150} = -\frac{1}{3} \]

Step 4: Interpret the Solution

The negative time value means there was an error in the setup or calculation step. Reassess initial conditions to ensure no logical errors in defining the situation.

Key Concepts

Flight Instruction

Flight instruction is a significant part of learning to fly, involving hands-on experience with actual aircraft. In this exercise, the cost per hour for flight instruction is given as $105. Learning in a real plane is indispensable because it helps pilots develop skills and confidence in handling the aircraft.
One key point to note is that Hai-Ling spent more time in flight instruction than in simulator training by 4 hours. This time difference plays a crucial role in how we set up the equations. Real-world flight experience can be costly due to aircraft rental, fuel costs, and the instructor’s fees, hence pinpointing the exact hours spent can help understand the financial investment in aviation training.

Simulator Training

Simulator training is an essential complement to flight instruction. It is a cost-effective way for pilots to practice procedures without being airborne. Each hour spent in the simulator in this problem costs $45.
Simulators are vital for developing skills in a safe environment, allowing pilots to practice emergency scenarios and improve their reaction times. Despite being less expensive, this mode of training is crucial for reinforcing theoretical knowledge practically. In this scenario, Hai-Ling has spent fewer hours in simulator training.

• This cost difference impacts your total training expense calculations.
• Understanding simulator versus flight hours is necessary to solve for total times spent in each area of training.

Variable Definition

Variables are symbols used to represent numbers or values in a mathematical situation. Here in your problem, defining variables correctly is crucial to solving the equations correctly.
Let's break it down:

• - Let \( x \) be the number of hours spent in the simulator.
• - Since flight instruction hours exceed simulator hours by 4, it is expressed as \( x+4 \).

This setup helps to place real-life concepts into mathematical frameworks. Properly understanding and defining these variables ensures that every aspect of the problem is accounted for in the calculations.

Error Correction

In mathematical problems, arriving at a negative value for time or other variables can indicate a need for error correction. This usually signifies an error in your equation or setup.
In Hai-Ling's scenario, upon finding \( x = -\frac{1}{3} \) hours for simulator time, it's clear there's a logical inconsistency, as time can't be negative.

• Reexamine variable assignments to ensure they adhere to the physical realities — time should always be positive.
• Recheck the step-by-step calculations, ensuring any errors in algebraic operations are addressed, especially combining like terms and arithmetic steps.

Checking the initial conditions, logical reasoning, and equations thoroughly can prevent errors, helping to recalibrate the calculations for correct outcomes.