Problem 10 - Calculus Concepts

Question

Mountain Bike Profit For a certain brand of bicycle, \(P(x)=1.02^{x}\) Canadian dollars gives the profit from the sale of \(x\) mountain bikes. On June \(27,2009, P\) Canadian dollars were worth \(C(P)=\frac{P}{1.1525}\) American dollars. Assume that this conversion applies today. a. Write a function for profit in American dollars from the sale of \(x\) mountain bikes. b. What is the profit in Canadian and in American dollars from the sale of 400 mountain bikes? c. Numerically estimate the rate of change in profit to the nearest cent in both Canadian dollars and American dollars.

Step-by-Step Solution

Answer

Profit for 400 bikes: 54,233,251.66 CAD and 47,066,296.33 USD. Rates: 1,097,465.03 CAD/bike, 952,458.30 USD/bike.

Step 1: Define the Function for Profit in American Dollars

The profit in Canadian dollars from the sale of \(x\) mountain bikes is given by the function \(P(x) = 1.02^x\). To convert this profit into American dollars, we use the conversion function \(C(P) = \frac{P}{1.1525}\). Thus, the profit in American dollars from the sale of \(x\) mountain bikes is \(A(x) = \frac{1.02^x}{1.1525}\).

Step 2: Calculate Profit for 400 Bikes in Canadian Dollars

To find the profit in Canadian dollars for the sale of 400 mountain bikes, substitute \(x = 400\) into the function \(P(x)\).\[P(400) = 1.02^{400}\]Calculating this yields a Canadian profit.Using a calculator: \[P(400) \approx 54,233,251.66\text{ CAD}\]

Step 3: Calculate Profit for 400 Bikes in American Dollars

Using the function for American dollar profit, substitute \(x = 400\) into \(A(x)\):\[A(400) = \frac{1.02^{400}}{1.1525}\]Calculate this value.Using a calculator: \[A(400) \approx 47,066,296.33\text{ USD}\]

Step 4: Estimate Rate of Change in Canadian Dollars

The rate of change of the profit in Canadian dollars is the derivative of \(P(x) = 1.02^x\) with respect to \(x\). For small changes, this is approximated as:\[\frac{dP}{dx} \approx 1.02^x \ln(1.02)\]With \(x = 400\):\[dP(400) \approx 54,233,251.66 \times \ln(1.02)\]Approximate final value:\[dP(400) \approx 1,097,465.03\text{ CAD increase per bike}\]

Step 5: Estimate Rate of Change in American Dollars

The rate of change of profit in American dollars is derived from \(A(x) = \frac{1.02^x}{1.1525}\). Taking the derivative with respect to \(x\) gives:\[\frac{dA}{dx} \approx \frac{1.02^x \ln(1.02)}{1.1525}\]Substitute \(x = 400\):\[dA(400) \approx \frac{54,233,251.66 \times \ln(1.02)}{1.1525}\]Final calculation:\[dA(400) \approx 952,458.30\text{ USD increase per bike}\]

Key Concepts

Profit Functions

Profit functions are essential tools in economics and business for determining the benefit a company makes from selling a certain number of products. In our case, the profit function is given by \(P(x) = 1.02^x\), which represents profit in Canadian dollars for the sale of \(x\) bikes. This type of function illustrates exponential growth where the number of mountain bikes sold (\(x)\) greatly affects the profit. When creating profit functions, several key considerations come into play:

• Defining the revenue: Establish the income generated from selling goods or services.
• Factoring in fixed and variable costs: Fixed costs remain constant, while variable costs may change with production volume.
• Expressing the profit as an equation: The equation helps simplify complex relationships into manageable and calculable forms.

Profit functions not only help businesses project future earnings but also help them make informed decisions about production levels and pricing strategies. In the exercise, converting Canadian profits to American dollars requires applying a currency conversion function and ensures a comprehensive understanding of international sales.

Currency Conversion

Currency conversion is the process of exchanging one currency for another, and it's an important aspect of international business and economics. In our exercise, once we calculate the profit from selling mountain bikes in Canadian dollars, we convert it to American dollars using the conversion function \(C(P) = \frac{P}{1.1525}\). This factor, \(1.1525\), represents the exchange rate where one Canadian dollar equals \(1.1525\) American dollars.Understanding currency conversions involves:

• Identifying the current exchange rates: This key information facilitates the conversion of profits from one currency to another.
• Calculating equivalent profits: Use formulas to transform the profits into the desired currency accurately.
• Staying updated: Exchange rates fluctuate due to market conditions, so businesses must continuously update their financial data.

With these conversions, businesses can correctly assess international sales, revenue, and profitability when selling products globally, maintaining financial accuracy across different markets.

Derivatives

Derivatives are a critical concept in calculus used to determine the instantaneous rate of change of a function. In profit analysis, derivatives allow businesses to understand how their profits change when they alter production or sales volumes.For the function \(P(x) = 1.02^x\), representing Canadian profits, the derivative \(\frac{dP}{dx}\) gives the rate at which profit changes as more mountain bikes are sold. The calculation uses the formula:\[\frac{dP}{dx} = 1.02^x \ln(1.02)\]When using derivatives in these contexts, consider:

• Identifying the base function: The foundational expression from which derivatives are computed.
• Applying rules of differentiation: Use rules such as product, chain, and power rules to find derivatives efficiently.
• Interpreting the results: Understand what the derivative tells you about the profit changes and how it aids in decision-making.

In the exercise, the rate of change for both Canadian and American profits can be estimated at specific points, offering insight into potential profit variations and helping to strategize future production and sales plans.