Problem 94 - Algebra and Trigonometry

Question

Manufacturing Cost A toy maker finds that it costs \(C=450+3.75 x\) dollars to manufacture \(x\) toy trucks. If the budget allows \(\$ 3600\) in costs, how many trucks can be made?

Step-by-Step Solution

Answer

The toy maker can manufacture 840 trucks.

Step 1: Identify the Equation

We are given the cost equation for manufacturing toy trucks as \(C = 450 + 3.75x\), where \(C\) is the total cost in dollars and \(x\) is the number of toy trucks. The problem states that the total budget for costs is \(\$3600\).

Step 2: Set up the Equation for Budget Constraint

We need to set the total cost equation equal to the budget limit. Therefore, we have: \[ 450 + 3.75x = 3600 \]

Step 3: Isolate the Variable

Subtract 450 from both sides of the equation to isolate the term with \(x\): \[ 3.75x = 3600 - 450 \] Simplifying the right side, we get: \[ 3.75x = 3150 \]

Step 4: Solve for the Number of Trucks

Divide both sides by 3.75 to find the value of \(x\): \[ x = \frac{3150}{3.75} \] Calculating the division, we find \(x = 840\).

Step 5: Verify the Solution

Plug \(x = 840\) back into the initial equation to verify: \(C = 450 + 3.75 \times 840 = 3600\), confirming the calculation is correct.

Key Concepts

Cost Functions

A cost function is a fundamental mathematical expression used to understand how costs vary with different levels of production, in this case, the number of toy trucks. It provides a way to model or predict the expenses associated with manufacturing products.
In our example, the cost function is given by the equation: - \(C = 450 + 3.75x\) - where \(C\) represents the total cost in dollars, and \(x\) denotes the number of toy trucks produced.

• The constant term, 450, represents fixed costs. These are expenses that do not change regardless of how many trucks are made, like rent and equipment costs.
• The term \(3.75x\) is the variable cost associated with producing each toy truck. If more trucks are produced, this cost increases proportionally.

Understanding this cost function allows us to predict how costs will change as production levels change, which is crucial for budgeting and planning in manufacturing.

Budget Constraints

Budget constraints are limitations on the spending ability of a company or an individual. In simple terms, it refers to how much money can be spent without exceeding available resources. For our toy manufacturing scenario, the budget constraint is set at \(\\(3600\). This means:

• The total manufacturing cost of toy trucks must not exceed \\)3600, aligning with the firm's financial capacity.
• Our task is to determine how many trucks can be made without breaching this budget limit.

To apply this constraint, we equate the cost function to our budget using the equation: \[450 + 3.75x = 3600\]Solving this equation lets us find out the maximum number of trucks that can be produced given the budget constraint.

Solving for Variables

Solving for variables is the process of finding the numerical value of unknowns in an equation. It involves manipulating the equation to isolate the variable on one side.In this problem, the goal is to find out how many toy trucks (\(x\)) can be produced under the budget constraint of \$3600. We start with the budget constraint equation: \[450 + 3.75x = 3600\]Here's how we solve it:- Subtract the fixed cost (450) from both sides to simplify: \[3.75x = 3600 - 450\]- This simplification gives us: \[3.75x = 3150\]The final step is to solve for \(x\) by dividing both sides by 3.75:\[x = \frac{3150}{3.75}\]Through this calculation, we determine that \(x = 840\). This means 840 toy trucks can be produced without exceeding the budget limit. Checking the solution by substituting \(x = 840\) back into the original cost function confirms our calculations, affirming our understanding of the steps involved.