Problem 60 - Calculus with Concepts in Calculus

Question

The Tee-rific Company produces 100,000 golf tees daily and sells them for \(5 \phi\) apiece. Assume that the total cost of producing one tee is \(2 \phi .\) Find the company's profit \(P\) (in cents) in terms of the number of working days.

Step-by-Step Solution

Answer

The profit \( P \) in cents is \( 300,000d \), where \( d \) is the number of working days.

Step 1: Understand the problem

The Tee-rific Company produces and sells 100,000 golf tees daily. Each tee is sold for 5 cents and costs 2 cents to produce. We are asked to find the profit in terms of the number of working days.

Step 2: Calculate revenue per day

Revenue per day is calculated by multiplying the number of tees sold in a day by the selling price per tee. Therefore, the daily revenue is: \( 100,000 \times 5 = 500,000 \) cents.

Step 3: Calculate cost per day

Cost per day is calculated by multiplying the number of tees produced in a day by the production cost per tee. Therefore, the daily cost is: \( 100,000 \times 2 = 200,000 \) cents.

Step 4: Determine daily profit

Profit for one day is the daily revenue minus the daily cost. Thus, the daily profit is: \( 500,000 - 200,000 = 300,000 \) cents.

Step 5: Express profit in terms of working days

Let \( d \) represent the number of working days. The total profit \( P \) over \( d \) days is given by: \( P = 300,000 \times d \) cents.

Key Concepts

Revenue Calculation

In the context of business, revenue calculation is a fundamental aspect. It involves determining the total income generated from selling goods or services. For the Tee-rific Company, this means understanding how much money they bring in from selling golf tees each day.
To calculate daily revenue, we multiply the number of items sold by the price of each item. In this case, Tee-rific sells 100,000 golf tees every day, with each tee priced at 5 cents. Therefore, the formula for revenue is:

• Number of tees sold: 100,000
• Price per tee: 5 cents

Using the formula, daily revenue = 100,000 tees \( \times \) 5 cents/tee = 500,000 cents. This calculation gives the company a clear picture of the income generated on a daily basis, an essential figure for further financial analysis.

Cost Calculation

Understanding cost calculation is crucial for determining how much it costs a company to produce its goods. At Tee-rific Company, we consider the expenses involved in manufacturing one golf tee.
The cost calculation is straightforward. By multiplying the number of tees produced by the cost per tee, we find the daily total production cost. For each tee, Tee-rific incurs a cost of 2 cents. Therefore, the daily cost calculation involves:

• Number of tees produced: 100,000
• Cost per tee: 2 cents

So, the daily production cost = 100,000 tees \( \times \) 2 cents/tee = 200,000 cents. This step ensures the company understands its expense commitment, serving as a foundation for profit calculation.

Daily Operations

Daily operations play a significant role in determining the overall profit. For Tee-rific Company, these operations include the production and sale of 100,000 golf tees each day. Efficient management of daily processes ensures the company maximizes its profit.
To find the daily profit, it's essential to subtract the total daily costs from the total daily revenue. Here, the daily revenue is 500,000 cents, and the daily cost is 200,000 cents, leading to:

• Daily revenue: 500,000 cents
• Daily cost: 200,000 cents
• Daily profit = Daily revenue - Daily cost = 500,000 - 200,000

Thus, the company's daily profit amounts to 300,000 cents. To express profit with respect to working days, if the company operates for \( d \) days, the total profit \( P \) is \( 300,000 \times d \) cents. Hence, by maintaining a streamlined daily operation, Tee-rific can effectively calculate its profits over any given period of time.