Problem 14 - Applied Calculus

Question

Give the cost, revenue, and profit functions. A lemonade stand operator sets up the stand for free in front of the neighbor's house, makes 5 quarts of lemonade for $$ 4,$ then sells each 8 -oz cup for 25 cents.

Step-by-Step Solution

Answer

Cost: \( C(q) = 4 \), Revenue: \( R(q) = 0.25q \), Profit: \( P(q) = 0.25q - 4 \).

Step 1: Define the Variables

First, we need to define the variables. Let \( q \) represent the number of 8-oz cups sold.

Step 2: Determine the Cost Function

The cost to make lemonade is given as $4 for 5 quarts. There is no additional cost for the stand. Since each quart contains 32 oz, 5 quarts yield 160 oz, meaning \( \frac{160}{8} = 20 \) cups. Thus, the cost function is constant: \( C(q) = 4 \), since all cups are included in this cost.

Step 3: Determine the Revenue Function

Each 8-oz cup is sold for 25 cents, so the revenue function based on \( q \) cups sold is \( R(q) = 0.25q \).

Step 4: Determine the Profit Function

Profit is calculated as revenue minus cost. Thus, the profit function is \( P(q) = R(q) - C(q) = 0.25q - 4 \).

Key Concepts

Cost Function

Understanding the cost function is a crucial part of any business, as it helps to determine how much is being spent to produce goods or services. In the case of the lemonade stand, we are told that the cost to make the lemonade is $4 for 5 quarts. One important thing to note here is that there isn't any additional cost associated with running the lemonade stand itself, such as rental fees or electricity costs, which is sometimes the case for more complex businesses.

To break it down further, we know that each quart of lemonade contains 32 ounces. Therefore, making 5 quarts gives a total of 160 ounces. Since each cup sold contains 8 ounces, we end up with 20 cups made from these 5 quarts:

• Total ounces = 5 quarts x 32 ounces/quart = 160 ounces
• Number of cups = 160 ounces ÷ 8 ounces/cup = 20 cups

Thus, the cost function, denoted as \( C(q) \), is constant and equal to 4, regardless of the number of cups sold (as long as it is within the 20 cups made from 5 quarts). This means that \( C(q) = 4 \) for \( q \leq 20 \). The cost function helps us recognize how the fixed cost in this scenario does not change with the number of units sold, offering a simplified view of the business's expenses.

Revenue Function

The revenue function reflects how much money is made from selling products or services, and is a key element in evaluating a business's earnings. For the lemonade stand, each 8-ounce cup is sold for 25 cents. To determine the total revenue from selling \( q \) cups, we use the following formula: \( R(q) = 0.25q \). This equation tells us that the revenue increases by 0.25 for each additional cup sold.

Here's how you can visualize it: Each time a cup is sold, the price of that cup, 0.25 dollars, contributes to the total revenue. For example:

• If 10 cups are sold, the total revenue would be \( R(10) = 0.25 \times 10 = 2.5 \) dollars.
• For 20 cups, the revenue would be \( R(20) = 0.25 \times 20 = 5 \) dollars.

This linear revenue function clearly shows how dependent revenue is on the quantity of product sold. This understanding allows businesses to forecast income based on their sales volume and strategize more effectively.

Profit Function

A profit function shows the difference between what is earned and what is spent. For the lemonade stand, the profit is calculated by subtracting the total costs from the total revenue. The formula for the profit function is \( P(q) = R(q) - C(q) \). For the lemonade stand, this becomes \( P(q) = 0.25q - 4 \).

Let's break it down: This function tells us that the profit for each additional cup sold increases by 0.25 dollars, but you'll need to sell enough cups to cover the initial cost of $4 before you actually start making a profit. This is where understanding the break-even point becomes essential.

• To find the break-even point, set \( P(q) = 0 \) and solve for \( q \):
• \( 0.25q - 4 = 0 \).
• \( 0.25q = 4 \).
• \( q = 16 \).

This result implies that you need to sell at least 16 cups just to break even. Every cup sold beyond this will contribute to profit. Understanding the profit function allows businesses to strategize prices and production levels to ensure they are financially sustainable.