Problem 13 - Intermediate Algebra

Question

The B737-400 aircraft costs \(\$ 2948\) dollars per hour to operate. The algebraic expression \(2948 t\) gives the total cost to operate the aircraft for \(t\) hours. Find the total cost to operate the \(\mathrm{B} 737-400\) for 3.6 hours.

Step-by-Step Solution

Answer

The total cost is $10,612.80.

Step 1: Understand the Expression

The expression given is \( 2948t \), where \( t \) represents the number of hours the aircraft is operated. \( 2948 \) is the cost per hour in dollars.

Step 2: Substitute the Hours

Substitute \( t = 3.6 \) into the expression. This means replacing \( t \) with 3.6 in the expression \( 2948t \).

Step 3: Calculate the Total Cost

Now, calculate \( 2948 \times 3.6 \). This involves multiplying the hourly cost by the number of hours to find the total operating cost.

Step 4: Perform the Multiplication

Perform the multiplication: \( 2948 \times 3.6 = 10612.8 \). This is the total cost to operate the B737-400 for 3.6 hours.

Key Concepts

Multiplication

Multiplication is a fundamental mathematical operation that combines groups of equal sizes. In the context of algebraic expressions, multiplication helps calculate unknown total values by acting as a bridge between a unit rate and its corresponding total quantity.

To understand multiplication within an algebraic expression, consider this simple example of multiplying a constant by a variable. In the expression \(2948t\), the number 2948 is constant, meaning it does not change. This is the cost per hour in dollars to operate an aircraft. The letter \(t\) represents the hours, which is the variable.

When we multiply these together, it signifies multiplying 2948 dollars for the number of hours the aircraft is in use.

Substitution

Substitution in algebra involves replacing a variable in an expression with a specific value to simplify or calculate the result. This concept is crucial for analyzing how mathematical expressions adapt to different scenarios.

When given the algebraic expression \(2948t\), if you know the specific number of hours the aircraft will be operational, you'll want to substitute this hour value into the expression to determine the total cost. For example, if \(t = 3.6\), you replace \(t\) in the expression with 3.6. Doing so converts \(2948t\) to \(2948 \times 3.6\), reflecting the operation of the aircraft for a particular timeframe.

Substitution allows us to evaluate algebraic expressions and gain concrete results for specific situations.

Cost Calculation

Calculating costs is often essential in real-world applications, especially in budgeting and financial planning. In our aircraft operation scenario, we use a defined algebraic expression to determine total expenses based on the duration of operation.

Once you substitute the time value in the expression, performing the cost calculation is straightforward. By multiplying the hourly cost and the hours (\(2948 \times 3.6\)), you find the total cost to operate the aircraft for that period. Hence, \(2948 \times 3.6 = 10612.8\) dollars.

This technique helps businesses and individuals estimate expenses reliably, allowing for efficient use of resources and better decision-making.