Problem 13 - Applied Calculus

Question

Give the cost, revenue, and profit functions. A couple running a house-cleaning business invests $$ 5000$ in equipment, and they spend $$ 15 in supplies to clean a house, for which they charge $$ 60 .

Step-by-Step Solution

Answer

Cost: \( C(x) = 5000 + 15x \); Revenue: \( R(x) = 60x \); Profit: \( P(x) = 45x - 5000 \).

Step 1: Define the Cost Function

The cost function, denoted by \( C(x) \), represents the total cost for cleaning \( x \) number of houses. It includes both fixed and variable costs.- The fixed cost is the initial investment of \\(5000 in equipment.- The variable cost is \\)15 per house.Therefore, the cost function is \( C(x) = 5000 + 15x \).

Step 2: Define the Revenue Function

The revenue function, denoted by \( R(x) \), describes the total revenue earned from cleaning \( x \) houses.- The cleaning fee for each house is \$60.Therefore, the revenue function is \( R(x) = 60x \).

Step 3: Define the Profit Function

The profit function, denoted by \( P(x) \), is the difference between the revenue and the cost for \( x \) houses.- Use the formula: \( P(x) = R(x) - C(x) \).- Substitute the expressions for \( R(x) \) and \( C(x) \).Thus, \( P(x) = 60x - (5000 + 15x) = 60x - 5000 - 15x \).Simplify to get: \( P(x) = 45x - 5000 \).

Key Concepts

Understanding the Cost Function

The cost function is a crucial starting point in any business as it helps determine how much is spent to provide a service or produce a product. In business calculus, a cost function, denoted by \( C(x) \), includes all the expenses associated with running a business.
There are two types of costs:

• **Fixed Costs:** These do not change based on the volume of goods or services produced. In our example, the \( 5000 \) investment in cleaning equipment is a fixed cost.
• **Variable Costs:** These vary with the number of houses cleaned or products made. Here, a cost of \( 15 \) per house for supplies is variable.

Hence, the formula for the cost function is constructed by summing fixed and variable costs: \[ C(x) = 5000 + 15x \].
This means for every house cleaned, the business spends \( 15 \), along with the initial equipment investment.

Defining the Revenue Function

Revenue functions are equally important as they help anticipate earnings. For our house-cleaning business, the revenue function, \( R(x) \), is straightforward.
It represents the amount of money collected from providing services. In this scenario, each clean generates \( 60 \).

• The revenue is strictly dependent on the number of houses cleaned.

Therefore, the formula for the revenue function is: \[ R(x) = 60x \].
This implies that for every house cleaned, the business earns \( 60 \). Calculating the revenue helps in understanding how sales contribute to covering costs and reaching profit goals.

Calculating the Profit Function

The profit function is what most businesses aim to maximize. It represents how much money remains after all costs are deducted from revenues. In mathematical terms, the profit function, \( P(x) \), is defined as:

• Profit (\( P(x) \)) = Revenue (\( R(x) \)) - Cost (\( C(x) \))

By substituting the earlier defined equations for revenue and cost, we have:
\[ P(x) = 60x - (5000 + 15x) \].
Breaking this down shows:

• The term \( 60x \) represents income generated per house.
• The expression \( (5000 + 15x) \) accounts for all expenses, including initial equipment costs and supplies per house.

Upon simplifying, we reach: \[ P(x) = 45x - 5000 \].
This indicates profit depends not only on the number of houses cleaned but also that a minimum of \( 112 \) houses must be cleaned before breaking even (i.e., when profit becomes positive).

• Break-even occurs where \( 45x = 5000 \), leading to approximately \( x = 111.11 \), rounded up to 112 houses.

Understanding the profit function helps businesses plan pricing, control costs, and strategize for growth.