Problem 29 - Applied Calculus

Question

A company produces and sells shirts. The fixed costs are 7000 dollars and the variable costs are 5 dollars per shirt. (a) Shirts are sold for 12 dollars each. Find cost and revenue as functions of the quantity of shirts, \(q\) (b) The company is considering changing the selling price of the shirts. Demand is \(q=2000-40 p\) where \(p\) is price in dollars and \(q\) is the number of shirts. What quantity is sold at the current price of $$ 12 ?\( What profit is realized at this price? (c) Use the demand equation to write cost and revenue as functions of the price, \)p .$ Then write profit as a function of price. (d) Graph profit against price. Find the price that maximizes profits. What is this profit?

Step-by-Step Solution

Answer

(a) Cost: \( C(q) = 7000 + 5q \), Revenue: \( R(q) = 12q \). (b) Quantity sold at $12: 1520 shirts, Profit: $3640. (c) Profit as function of price: \( P(p) = -40p^2 + 2200p - 17000 \). (d) Max profit at $27.5 per shirt is $18500.

Step 1: Define Cost and Revenue Functions

The cost function, which includes fixed and variable costs, is given by \( C(q) = 7000 + 5q \). The revenue from selling shirts at $12 each is given by \( R(q) = 12q \). These are linear functions of the quantity \( q \).

Step 2: Calculate Quantity Sold at Current Price

Using the demand equation \( q = 2000 - 40p \), substitute \( p = 12 \) to find the current quantity sold. \( q = 2000 - 40(12) = 2000 - 480 = 1520 \). So, 1520 shirts are sold.

Step 3: Determine Profit at Current Price

Profit is calculated as revenue minus cost, \( P(q) = R(q) - C(q) \). Substituting values at \( q = 1520 \), we have \( R(1520) = 12 \times 1520 = 18240 \) and \( C(1520) = 7000 + 5 \times 1520 = 14600 \). Then, \( P(1520) = 18240 - 14600 = 3640 \).

Step 4: Write Cost, Revenue, Profit as Functions of Price

Express \( q \) from the demand equation in terms of \( p \): \( q = 2000 - 40p \). Substituting into the cost: \( C(p) = 7000 + 5(2000 - 40p) = 17000 - 200p \). Revenue in terms of price: \( R(p) = p(2000 - 40p) = 2000p - 40p^2 \). Profit: \( P(p) = R(p) - C(p) = 2000p - 40p^2 - (17000 - 200p) \). Simplifying, \( P(p) = 2200p - 40p^2 - 17000 \).

Step 5: Maximize Profit by Finding Vertex of Quadratic

The profit function \( P(p) = -40p^2 + 2200p - 17000 \) is quadratic, opening downwards. Maximum profit occurs at the vertex. Vertex formula: \( p = -\frac{b}{2a} = -\frac{2200}{2(-40)} = 27.5 \). Substitute to find maximum profit: \( P(27.5) = -40(27.5)^2 + 2200(27.5) - 17000 = 18500 \).

Step 6: Define and Interpret the Graph of Profit by Price

The graph of the profit function \( P(p) = -40p^2 + 2200p - 17000 \) is a downward-opening parabola. The vertex \( p = 27.5 \) represents the price for maximum profit, which is \( 18500 \). This confirms that selling price should be about $27.5 per shirt to maximize profit.

Key Concepts

Revenue Function

The revenue function helps us understand how much money a company makes from selling its products. For this exercise, the revenue function is given by the formula \( R(q) = 12q \), where \( q \) represents the quantity of shirts sold. Revenue is essentially the product of the number sold and the selling price per unit. In this case, since each shirt is sold for $12, to find out how much the company earns altogether, we multiply the price by the total number of shirts sold.
Think of it this way: the more shirts sold, the higher the revenue, assuming the price remains constant. This linear relationship is key in revenue calculations. Linear functions like this make it easy to quickly see how changes in sales directly impact revenue.
In practice, businesses monitor their revenue function to ensure they're selling enough quantity to cover costs and ideally make a profit.

Cost Function

The cost function gives insight into the total expenses a company incurs in the production process. Here, the cost function is expressed as \( C(q) = 7000 + 5q \), where \( q \) is the number of shirts produced and sold. The cost comprises fixed costs, which are constant regardless of quantity, and variable costs that vary with production levels.
Fixed costs are like the baseline expenses required just to keep the business running, amounting to \(7000 in this context. Variable costs, on the other hand, increase with each additional unit produced, here \)5 per shirt. This combination gives us the total cost, reflecting both stable and fluctuating expense factors.
This understanding helps businesses plan and predict future expenses, crucial for making strategic decisions about pricing and production volumes. Managing the cost function effectively can lead to better profitability outcomes.

Demand Equation

The demand equation illustrates how many shirts are sold depending on the shirt price. It is given as \( q = 2000 - 40p \), with \( q \) representing the quantity demanded and \( p \) the price per shirt in dollars. This equation shows an inverse relationship between price and demand: as the price increases, the demand tends to decrease.
When the shirts are priced at $12 each, substituting into the equation gives \( q = 2000 - 40 \times 12 = 1520 \), meaning 1520 shirts are sold. This relationship is crucial for businesses as it informs them how changes in pricing affect sales volume.
Understanding demand elasticity, which refers to how sensitive demand is to price changes, helps businesses set prices strategically. When demand is elastic, a small change in price could lead to a significant change in quantity sold.

Quadratic Function Analysis

Quadratic function analysis is applied to find the price point that maximizes profit. The profit function derived from costs and revenue in terms of price is \( P(p) = -40p^2 + 2200p - 17000 \). This is a quadratic function, characterized by its parabolic shape which opens downward due to the negative coefficient of \( p^2 \).
The maximum profit is located at the vertex of this parabola. Using the vertex formula, \( p = -\frac{b}{2a} \), we compute the optimal price to be \( p = 27.5 \). At this price, the maximum profit is calculated to be 18500 dollars.
This analysis is vital in business as it enables finding the balance between price and quantity to achieve the highest profit. Understanding and applying quadratic function analysis can lead businesses to optimal pricing strategies, ensuring competitive advantage and profitability.