Problem 34 - Applied Calculus

Question

A taxi company has an annual budget of 720,000 dollars to spend on drivers and car replacement. Drivers cost the company 30,000 dollars each and car replacements cost 20,000 dollars each. (a) What is the company's budget constraint equation? Let \(d\) be the number of drivers paid and \(c\) be the number of cars replaced. (b) Find and interpret both intercepts of the graph of the equation.

Step-by-Step Solution

Answer

(a) \(30,000d + 20,000c = 720,000\); (b) Intercepts: Drivers (24,0), Cars (0,36).

Step 1: Identify Variables and Costs

Identify the variables for the exercise: let \(d\) represent the number of drivers and \(c\) represent the number of cars replaced. The cost per driver is \(30,000\) dollars, and the cost per car is \(20,000\) dollars.

Step 2: Construct the Budget Constraint Equation

The company can spend a total of 720,000 dollars on drivers and cars. So, the equation that represents this budget constraint is \(30,000d + 20,000c = 720,000\).

Step 3: Find the Drivers Intercept

To find the intercept on the \(d\)-axis, set \(c = 0\) in the equation. This gives us \(30,000d = 720,000\). Solving for \(d\), we have \(d = \frac{720,000}{30,000} = 24\). Thus, the \(d\)-intercept is (24, 0), meaning if no cars are replaced, the company can hire 24 drivers.

Step 4: Find the Cars Intercept

To find the intercept on the \(c\)-axis, set \(d = 0\) in the equation. This results in \(20,000c = 720,000\). Solving for \(c\), we have \(c = \frac{720,000}{20,000} = 36\). Therefore, the \(c\)-intercept is (0, 36), implying if no drivers are paid, the company can replace 36 cars.

Key Concepts

Understanding Linear Equations in Budgeting

In the context of budget management, linear equations are incredibly useful tools. They allow us to represent financial constraints and scenarios in a simple mathematical form. In this scenario, a taxi company must juggle how many drivers they can hire and how many cars they can replace within their annual budget of $720,000. This situation is captured by a linear equation with two variables: the number of drivers, denoted as \( d \), and the number of cars replaced, denoted as \( c \). The budget constraint can be described as \( 30,000d + 20,000c = 720,000 \). This equation is linear because it forms a straight line when plotted on a graph. Each point on this line represents a possible combination of drivers and car replacements that maximizes, but does not exceed, the budget. Calculating the different possible values for \( d \) and \( c \) helps a company understand how to balance their spending.

In practice, linear equations like this one are essential for businesses when allotting funds to various sectors while ensuring they do not exceed available resources. This makes them a crucial part of cost analysis and financial planning strategies.

Identifying and Interpreting Intercepts

Intercepts in linear equations offer significant insights into extreme scenarios of the budget constraint. In this problem, the \( d \)-intercept and \( c \)-intercept show us what happens when we allocate the entire budget to only one of the costs. The \( d \)-intercept is found by setting \( c = 0 \). Solving the equation \( 30,000d = 720,000 \), we find \( d = 24 \). This means if the company does not replace any cars, they can employ 24 drivers. In real-world terms, this highlights the maximum capacity for hiring if no other expenses are considered. Now for the \( c \)-intercept, set \( d = 0 \). From \( 20,000c = 720,000 \), we find \( c = 36 \). Hence, with no drivers paid, 36 cars can be replaced.

The intercepts provide a clear understanding of the operational limits while considering only one cost. They help visualize how cost distribution can be maximized in either direction, guiding strategic decision-making in resource allocation.

Exploring Cost Analysis Through Budget Constraints

Cost analysis is an integral part of financial decision-making, particularly when dealing with fixed budgets. The exercise involving a taxi company's budget illustrates how businesses allocate resources for optimal results. In cost analysis, understanding the implications of each unit cost — such as $30,000 per driver and $20,000 per car replacement — is vital. It helps determine how each financial decision impacts the overall budget. Breaking down the total cost in these units clarifies how money is spent and ensures that future expenditures are strategically justified.

Further, this exercise shows how a budget constraint, a concept rooted in cost analysis, acts as a tool to prevent overspending. It aids in mapping out various feasible scenarios (combinations of drivers hired and cars replaced) under financial restrictions. Strategically managing these limitations can enhance a company's operational efficiency and financial health. Businesses can better control their spending limits while navigating economic uncertainties by using budget constraints and cost analysis effectively.