Question

FREIGHT TRAINS A train with 150 freight cars is used to haul two types of grain. Each freight car can haul 97.3 tons of barley or 114 tons of corn. Let \(n\) represent the number of freight cars containing corn. Which function correctly represents the total weight the train can haul? $$ \text { A. } W=97.3(150-n)+114 n \quad \text { B. } W=97.3 n+114(150-n) $$

Short answer

The function that correctly represents the total weight the train can haul is A. \(W=97.3(150-n) + 114n\).

Step-by-step solution

Understand the variables in the function

In this scenario, \(n\) represents the number of freight cars containing corn, so \(150 - n\) will represent the number of freight cars carrying barley. This is because the total number of freight cars is given as 150. Thus, each freight car used for corn reduces the number of cars available for barley.

Use the payload of each grain type

Each freight car can convey 97.3 tons of barley or 114 tons of corn. Therefore, the total weight loaded with barley will be \(97.3 \times (150-n)\), and the total weight loaded with corn will be \(114 \times n\)

Select the correct function

Summing the two expressions from the previous step, the total weight \(W\) that the train can haul becomes: \(W=97.3(150-n) + 114n\). Therefore, the correct answer is option A.

Key concepts

Algebraic Expressions

Algebraic expressions are mathematical phrases made up of numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and division. They are fundamental ingredients in forming equations and can represent real-world problems in a compact and general form.

An example from the textbook exercise demonstrates this well. The total weight that a train can haul, with a mix of corn and barley across its freight cars, is expressed algebraically. For barley, you have the weight per car multiplied by the number of cars carrying barley, forming the expression \(97.3(150-n)\). The \(n\) represents the number of cars with corn, hence the remaining cars, expressed as \((150-n)\), must be carrying barley. Similarly, the weight for corn is \(114n\), representing \((114\) tons per car) multiplied by the number of cars with corn, \(n\). Combining these, you get the total weight \(W\) as an algebraic expression \(W=97.3(150-n) + 114n\).

When you see algebraic expressions, remember that they encapsulate relationships between variables and constants and can simplify or model complex situations with a simple formula.

Systems of Equations

When we deal with more than one unknown quantity, and we have multiple algebraic expressions to connect these quantities, we're looking at a system of equations. Systems of equations can have one or more solutions, which are the set of values for the variables that make all the equations true simultaneously.

In the case of our freight train, if we had another condition to meet—say a maximum weight the train can carry while still needing to transport a certain minimum of each grain—we would end up with multiple algebraic expressions that need to be solved together. While our current problem only required us to understand a single function, in the real world, often problems are multi-faceted and need systems of equations to encapsulate all conditions. Solving such systems requires techniques like graphing, substitution, elimination, or using matrices depending on the complexity and the number of equations involved.

Variables

Variables are symbols used to represent unspecified numbers or values. In algebra, they are the basic tools that allow us to describe and solve problems with unknown quantities. Variables can change and take on different values, which is why they are called 'variable'.

In our textbook exercise, the variable \(n\) is used to denote the number of freight cars containing corn. It's a placeholder that can represent any number between 0 and 150, as the train could potentially carry all corn, all barley, or any combination in between. The beauty of using a variable is that we don't need to know the exact number of corn-filled cars to understand and create a function that describes the total weight for any configuration of the freight load. This technique of using variables is crucial because it provides flexibility and adaptability in mathematical equations and expressions, allowing for general solutions rather than case-by-case calculations.