Problem 12 - Calculus an Applied Approach

Question

cost, Revenue, and Profit A company that manufactures pet toys calculates that its costs and revenue can be modeled by the equations \(C=75,000+1.05 x\) and \(R=500 x-\frac{x^{2}}{25}\) where \(x\) is the number of toys produced in 1 week. If production in one particular week is 5000 toys and is increasing at a rate of 250 toys per week, find: (a) the rate at which the cost is changing. (b) the rate at which the revenue is changing. (c) the rate at which the profit is changing.

Step-by-Step Solution

Answer

(a) The cost rate is \(1.05\) dollars per toy, (b) The revenue rate is \(300\) dollars per toy, (c) The profit rate for the week in question is \(298.95\) dollars per toy

Step 1: Compute the Cost rate

Differentiate the cost function with respect to \(x\) gives \(\frac{dC}{dx}=1.05\). This rate is constant and does not change with \(x\). This indicates the cost per toy is constant with the increase in number of toys.

Step 2: Compute the Revenue rate

Differentiate the revenue function with respect to \(x\) to find the rate of change. This gives, \(\frac{dR}{dx}=500-\frac{2x}{25}\) Now, Substitute \(x=5000\) in the above equation, that gives \(\frac{dR}{dx}=500-\frac{2(5000)}{25}=300\). This means, for a weekly increase of 250 toys, weekly revenue increase is \(300*250=75000$.

Step 3: Compute the Profit rate

Profit is defined as Revenue - Cost, Differentiate the profit function \(P=R-C\) with respect to \(x\) to find the rate of change. This gives \(\frac{dP}{dx}=\frac{dR}{dx}-\frac{dC}{dx} = 300-1.05 = 298.95\). This means, for a weekly increase of 250 toys, weekly profit increase is \(298.95*250=74737.5$

Key Concepts

Cost Rate Calculation

Understanding the cost rate calculation is essential when analyzing the financial performance of a business. In the provided exercise, the cost function, represented by the equation \( C=75,000+1.05x \), helps us assess the total costs associated with producing \( x \) number of pet toys in a given week.

To calculate the rate at which costs are changing, the differentiation of functions comes into play. By differentiating the cost function with respect to the number of toys produced \( x \), we obtain \( \frac{dC}{dx}=1.05 \). This result implies that the cost rate is constant, meaning for each additional toy produced, the cost increases by $1.05, regardless of the production level.

Importance in Business Decisions

This information is crucial for business as it allows managers to predict how increasing production will affect costs directly. A constant cost rate simplifies budget planning as it gives a linear relationship between the number of units produced and total cost.

Revenue Rate Calculation

Next, let's explore revenue rate calculation, which is a measure of how much money a company is bringing in through sales as production varies. In the context of our exercise, the revenue function is given by \( R=500x-\frac{x^2}{25} \) where \( x \) is the number of toys sold.

To find out how revenue is changing as production increases, we also turn to differentiation. Differentiating the revenue function yields \( \frac{dR}{dx}=500-\frac{2x}{25} \). Substituting \( x=5000 \) into this derivative, the revenue rate of change is \( 300 \) dollars per toy.

Strategic Pricing and Marketing

This rate of change can influence pricing strategies and marketing efforts. It is indicative of the revenue gained for each additional unit produced after reaching a production level of 5000 toys. For businesses, understanding how revenue changes with production is key to maximizing sales and profit.

Profit Rate Calculation

The ultimate goal of any business is to make a profit. Profit rate calculation offers insights into the efficiency of a company's operations. In our exercise, we have the cost and revenue functions, so profit \( P \) is simply the difference between them: \( P=R-C \).

The derivative of the profit function with respect to \( x \), \( \frac{dP}{dx} \), represents the rate at which profit is changing relative to the number of toys produced. From the exercise, we have \( \frac{dR}{dx}=300 \) and \( \frac{dC}{dx}=1.05 \), so the profit rate is \( 298.95 \).

Operational Efficiency and Growth

In business, knowing the profit rate is fundamental for making decisions about scaling production, investing in new equipment, or entering new markets. A positive profit rate that increases with production is a good indicator of a healthy and growing business venture.

Differentiation of Functions

At the heart of these calculations lies the process of differentiation, a cornerstone of calculus. Differentiation enables us to calculate the rate of change of a function; in business applications, these functions are often related to cost, revenue, and profit.

Through differentiation, we are able to find out how a slight change in the number of units produced (\( dx \)) will affect cost \( \left(\frac{dC}{dx}\right) \), revenue \( \left(\frac{dR}{dx}\right) \), and profit \( \left(\frac{dP}{dx}\right) \). This information is invaluable as it provides a snapshot of the dynamic environment of business operations.

Application in Real-World Scenarios

For instance, suppose demand spikes unexpectedly, and the company needs to determine whether it can meet this demand profitably. Utilizing the differentiation of these functions, it can predict additional costs and revenues, and ultimate impact on profit, for any potential increase in production volumes.