Problem 30 - Applied Calculus

Question

A taxi company has an annual budget of $$\$ 720,000$$ to spend on drivers and car replacement. Drivers cost the company $$\$ 30,000$$ each and car replacements cost $$\$ 20,000$$ cach. (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) Budget equation: \( 30,000d + 20,000c = 720,000 \). (b) Drivers intercept: 24, Cars intercept: 36.

Step 1: Formulate the Budget Constraint Equation

The company has a total budget of \(720,000, which is spent on drivers and car replacements. Let \( d \) be the number of drivers and \( c \) be the number of cars. Given that drivers cost \)30,000 each and cars cost $20,000 each, the budget constraint equation is:\[ 30,000d + 20,000c = 720,000 \]

Step 2: Solve for the Driver Intercept

To find the driver intercept, set the number of cars \( c = 0 \) and solve for \( d \):\[30,000d + 20,000 \times 0 = 720,000 \30,000d = 720,000 \d = \frac{720,000}{30,000} = 24\]This means if no cars are replaced, the company can pay 24 drivers.

Step 3: Solve for the Car Intercept

To find the car intercept, set the number of drivers \( d = 0 \) and solve for \( c \):\[30,000 \times 0 + 20,000c = 720,000 \20,000c = 720,000 \c = \frac{720,000}{20,000} = 36\]This means if no drivers are paid, the company can replace 36 cars.

Step 4: Interpret the Intercepts

The driver intercept indicates the maximum number of drivers (24) that can be paid if no cars are replaced. The car intercept indicates the maximum number of cars (36) that can be replaced if no drivers are paid. These intercepts show the trade-offs the company faces between employing drivers and replacing cars under its budget constraint.

Key Concepts

Linear Equation

In the world of budgeting, a linear equation is a helpful tool to understand how resources are allocated. For a taxi company, the budget constraint can be expressed as a linear equation. This type of equation combines the costs of multiple expense categories to reach a total budget limit. The general form for this particular application is: \[ 30,000d + 20,000c = 720,000 \]This equation illustrates the relationship between two variables, drivers and car replacements. It clearly defines the total expenditure by summing the products of the quantity and unit cost of each resource. Linear equations in budgeting are vital as they help firms visualize financial limits and guide optimal allocation of resources.

Intercepts

Intercepts are critical points on a graph where the line crosses the axes. In our taxi company example, there are two important intercepts:

• Driver Intercept: Found by setting car replacements \(c = 0\), resulting in \(d = 24\).
• Car Intercept: Found by setting drivers \(d = 0\), resulting in \(c = 36\).

Interpretation of intercepts shows extremes in resource allocation. For instance, a driver intercept of 24 means the company can employ 24 drivers if no cars are replaced. Similarly, a car intercept of 36 implies 36 cars can be replaced if no drivers are paid. These intercepts give valuable insight into maximum capacity at the boundaries of spending.

Trade-Offs in Budgeting

Every budgeting decision involves trade-offs, and this is starkly visible in our taxi company's budgeting scenario. The linear equation, \(30,000d + 20,000c = 720,000\), highlights these trade-offs. The company must choose how to allocate their budget between hiring drivers and replacing cars.

• If they spend more on drivers, they must spend less on cars.
• More spending on cars leaves less available for drivers.

This balancing act is a fundamental aspect of all budgeting processes. Understanding these trade-offs helps the company decide how to distribute resources efficiently to achieve their goals without overspending.

Graph Interpretation

Graph interpretation is an essential skill for those looking to manage budgets effectively. By plotting the linear equation, \(30,000d + 20,000c = 720,000\), on a graph, one can see how different allocations of resources affect the budget.

• The line represents all possible combinations of drivers and car replacements that fit within the budget.
• The slope of the line indicates the rate at which one resource must be reduced to increase another without changing total spending.
• Intercepts on the graph provide visual representations of maximum allocations.

Through graph interpretation, companies can better strategize and predict how various budgeting options will unfold. By seeing visual representations, stakeholders can more effectively understand and make informed decisions on resource allocation.