Problem 30 - Applied Calculus

Question

A health club has cost and revenue functions given by \(C=10,000+35 q\) and \(R=p q,\) where \(q\) is the number of annual club members and \(p\) is the price of a one year membership. The demand function for the club is \(q=3000-20 p\) (a) Use the demand function to write cost and revenue as functions of \(p\) (b) Graph cost and revenue as a function of \(p,\) on the same axes. (Note that price does not go above \(\$ 170\) and that the annual costs of running the club reach \(\$ 120,000 .)\) (c) Explain why the graph of the revenue function has the shape it does. (d) For what prices does the club make a profit? (e) Estimate the annual membership fee that maximizes profit. Mark this point on your graph.

Step-by-Step Solution

Answer

(a) Cost: \(C = 115,000 - 700p\), Revenue: \(R = 3000p - 20p^2\). (b) Graph functions. (c) Revenue is a downward parabola. (d) Prices for profit: \(p < \) about \(107.5\). (e) Maximum profit at \(p = \$92.5\).

Step 1: Express Demand as a Function of Price

Start with the demand function: \(q = 3000 - 20p\). This equation shows how the quantity of memberships \(q\) varies with price \(p\).

Step 2: Substitute into Cost Function

The cost function is given by \(C = 10,000 + 35q\). Substitute \(q = 3000 - 20p\) into the cost function:\[C = 10,000 + 35(3000 - 20p)\].

Step 3: Simplify Cost Function

Simplify the expression from Step 2 to find cost as a function of price \(p\):\[C = 10,000 + 105,000 - 700p = 115,000 - 700p\].

Step 4: Substitute into Revenue Function

The revenue function is given by \(R = pq\). Substitute \(q = 3000 - 20p\) into the revenue function:\[R = p(3000 - 20p) = 3000p - 20p^2\].

Step 5: Graph Cost and Revenue Functions

Plot the functions \(C = 115,000 - 700p\) and \(R = 3000p - 20p^2\) on the same set of axes with \(p\) on the x-axis. Ensure \(p\) does not exceed \(170 and recognize that \(C\) reaches \)120,000.

Step 6: Analyze Revenue Graph Shape

The revenue function \(R = 3000p - 20p^2\) is a quadratic function opening downwards (negative coefficient of \(p^2\)), creating a parabola which initially increases and then decreases as \(p\) increases.

Step 7: Determine Profit Conditions

Profit occurs when revenue is greater than cost: \(R > C\). Set up the inequality:\[3000p - 20p^2 > 115,000 - 700p\], and solve for \(p\), considering real solutions within the valid range of \(p\).

Step 8: Maximize Profit

To find the price that maximizes profit, take the derivative of the profit function \(P(p) = R(p) - C(p)\) and find the critical points. Where \(P'(p) = 0\), calculate the maximum by solving:\[3000 - 40p + 700 = 0\] to find \(p = 92.5\). Plot this on the graph.

Key Concepts

Cost Function

The cost function is an equation representing the total expenses of operating a business as a function of output quantity. In the health club example, the cost function is given by:\[C = 10,000 + 35q\].
Here, \(C\) is the total cost, \(10,000\) is the fixed cost, and \(35\) is the variable cost per member, represented by \(q\).
To express the cost as a function of price, we substitute \(q = 3000 - 20p\) from the demand function into the cost function:
- Substitute \(q\) into the cost function: \(C = 10,000 + 35(3000 - 20p)\)
- Simplify: \(C = 115,000 - 700p\)
This means the health club’s cost depends on the price of the membership, with a decrease in cost as membership price \(p\) increases.
Understanding the cost function helps businesses manage expenses and strategize pricing effectively. The relationship between cost and price is crucial for making informed decisions about pricing structures.

Revenue Function

The revenue function describes how much money a company earns through sales. In our exercise, the revenue function is \(R = pq\), where \(p\) is the price per membership and \(q\) is the demand, or the number of memberships sold.
Substituting the demand function \(q = 3000 - 20p\) into the revenue function, we have:- \(R = p(3000 - 20p)\)- Simplify to \(R = 3000p - 20p^2\)
The result is a quadratic equation, representing a parabolic curve when graphed. Initially, as the price \(p\) increases, revenue \(R\) also increases, reaching a peak. After this peak, further increases in price cause revenue to decrease. This occurs because higher prices reduce demand significantly.
Recognizing this pattern helps set optimal pricing strategies by identifying the price range that maximizes revenue before reaching a decrease due to low demand. The insights drawn from the revenue curve are essential for strategic business planning.

Demand Function

The demand function describes how the quantity of a product demanded by consumers changes with its price. The demand function in this context is \(q = 3000 - 20p\).
- Here, \(q\) is the number of memberships, and \(p\) is the price of a membership.
- As the price \(p\) increases, the demand \(q\) decreases linearly.
This linear relationship indicates that there's an inverse relationship between price and quantity demanded.
Understanding the demand function is vital for any business as it shows how changes in price affect consumer behavior. By analyzing the demand function, the health club can predict how many memberships it will sell at different price points. Knowing this can help optimize pricing strategies, enabling the business to set prices that attract the maximum number of customers while still achieving overall revenue and profit goals.
Calculating the intercepts in the demand function can give insights into maximum potential demand when the price is zero, and how demand drops as price increases. The information from the demand function helps balance between the lowest possible price and maintaining profitability.