Problem 43 - Calculus an Applied Approach

Question

Profit The demand function for a product is given by \(p=50 / \sqrt{x}\) for \(1 \leq x \leq 8000,\) and the cost function is given by \(C=0.5 x+500\) for \(0 \leq x \leq 8000\). Find the marginal profits for (a) \(x=900,\) (b) \(x=1600,\) (c) \(x=2500,\) and (d) \(x=3600\). If you were in charge of setting the price for this product, what price would you set? Explain your reasoning.

Step-by-Step Solution

Answer

The marginal profits for x=900, x=1600, x=2500 and x=3600 are 0.83, 0.625, 0.5 and 0.417, respectively. If looking to maximize marginal profit, the price should be set when x=900. Then, the price would be $5/3.

Step 1: Find the Revenue Function

The revenue \(r(x)\) is given by the product of price and demand i.e. \(r(x) = p * x = 50x / \sqrt{x}\ = 50\sqrt{x}\).

Step 2: Calculate the Marginal Revenue

The marginal revenue \(r'(x)\) is the derivative of the revenue function w.r.t. x. \(r'(x) = 25 / \sqrt{x}\).

Step 3: Determine the Marginal Cost

The marginal cost \(C'(x)\) is the derivative of the cost function. Therefore, \(C'(x) = 0.5\).

Step 4: Compute the Marginal Profit for x=900

The marginal profit for x=900 is the marginal revenue at x=900 minus the marginal cost, \(r'(900) - C'(900) = 25 / \sqrt{900} - 0.5 = 0.83\).

Step 5: Repeat for x=1600, x=2500 and x=3600

Repeat the procedure for x=1600, x=2500 and x=3600 to get the marginal profits for these values of x; For x=1600; \(r'(1600) - C'(1600) = 25 / \sqrt{1600} - 0.5 = 0.625\), for x=2500; \(r'(2500) - C'(2500) = 25 / \sqrt{2500} - 0.5 = 0.5\) and for x=3600 ; \(r'(3600) - C'(3600) = 25 / \sqrt{3600} - 0.5 = 0.417\).

Step 6: Find the Best Price

If looking for maximising the marginal profit, the best price should be set when x=900, as \(0.83\) is the highest value obtained. So, the price would be \(50 / \sqrt{900} = 5/3\).

Key Concepts

Demand Function

The demand function describes the relationship between the price of a product and the quantity of the product that consumers are willing to buy. In this scenario, the demand function is represented as \( p = \frac{50}{\sqrt{x}} \). This formula indicates that as the quantity \( x \) increases, the price \( p \) decreases.
Understanding this relationship is essential for setting prices and predicting consumer behavior.

• If \( x \), the quantity, is small, the price will be high because there's a low supply or high demand.
• As \( x \) increases, the availability increases, leading to a lower price because of higher supply or lower demand.

Cost Function

The cost function illustrates how total costs change with varying levels of production. In our case, it's given by \( C = 0.5x + 500 \). Here, \( x \) represents the quantity produced, \( 0.5x \) is a variable cost dependent on the number of units produced, and 500 is a fixed cost.
Understanding cost structures helps businesses make informed decisions about production levels and pricing strategies.

• Fixed costs, like the 500 here, remain constant regardless of the output level.
• Variable costs, on the other hand, will increase as production ramps up.

Revenue Function

Revenue is the income generated from selling goods or services and is obtained by multiplying the price by the quantity sold. The revenue function is noted as \( r(x) = 50\sqrt{x} \), computed from the product of demand, \( p(x) \), and quantity, \( x \).
Revenue is crucial because it reflects the company's ability to generate sales.

• A higher quantity \( x \) typically results in higher revenue, assuming the decrease in price is not faster than the increase in quantity sold.
• It's important to find the balance where both price and quantity positively affect total revenue.

Marginal Revenue

Marginal revenue is the additional revenue earned by selling one more unit of a product. It's given by the derivative of the revenue function. For our case, \( r'(x) = \frac{25}{\sqrt{x}} \).
Marginal revenue helps businesses decide where increasing production further is beneficial.

• If marginal revenue exceeds the cost of producing one more unit, increasing production could be profitable.
• When marginal revenue starts declining, it indicates reaching or passing the optimum production point.

Marginal Cost

Marginal cost is the cost of producing one additional unit. It can be calculated by taking the derivative of the cost function. Here, it is \( C'(x) = 0.5 \).
Knowing the marginal cost, businesses can assess how production changes affect overall profitability.

• Constant marginal costs, as seen here, suggest that each additional item costs the same amount to produce.
• Firms often aim to produce until marginal cost equals marginal revenue, maximizing profit.