Problem 12 - College Algebra and Calculus: An Applied Approach

Question

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

The rate at which cost is changing is $1.05 per toy. The rate at which revenue is changing is -$150 per toy. The profit is thus decreasing by $151.05 per toy.

Step 1: Differentiate the Cost Function

First, we differentiate the Cost equation \(C=75,000+1.05x\). The derivative, \(C'(x)\), will provide the rate at which cost is changing per toy. The constant 75,000 disappears when differentiated, and the derivative of \(1.05x\) is just 1.05.

Step 2: Evaluate the Cost's Rate of Change

We then substitute the number of toys, \(x=5000\), into the rate of change equation, \(C'(x)=1.05\). We find that the cost is increasing at a rate of $1.05 per toy.

Step 3: Differentiate the Revenue Function

We do the same for the Revenue function, \(R=500x-\frac{x^{2}}{25}\). First, we take its derivative which is \(R'(x)=500-\frac{2x}{25}\).

Step 4: Evaluate the Revenue's Rate of Change

Then we substitute \(x=5000\) into \(R'(x)=500-\frac{2x}{25}\). After simplifying, we find that revenue is changing at a rate of \(-150 per toy.\)

Step 5: Calculate the Profit's Rate of Change

To find the rate at which profit is changing, we subtract the cost's rate of change from the revenue's. In mathematical terms, the profit's rate of change, \(P'(x)\), equals the revenue's rate of change, \(R'(x)\), subtract the cost's rate, \(C'(x)\). So, \(P'(x) = R'(x) - C'(x)\). Substituting the values from before, we find the profit to be decreasing at a rate of $151.05 per toy.

Key Concepts

Cost Function

In business and economics, understanding the cost function is essential. A cost function describes the total cost of production as a function of the quantity produced. In the given exercise, the cost function is represented by the equation: \[ C = 75,000 + 1.05x \]Here, \( C \) is the total cost and \( x \) is the number of toys produced. The constant 75,000 represents fixed costs, which are costs that do not change regardless of the production quantity. The term \( 1.05x \) represents the variable cost, which increases linearly with the number of toys produced. This means for every additional toy produced, the cost goes up by \(1.05.To find the rate at which costs are changing as production increases, we differentiate the cost function with respect to \( x \). The resulting constant derivative, 1.05, tells us that costs increase by \)1.05 per toy produced. This information is crucial for budgeting and forecasting future spending.

Revenue Function

The revenue function gives us insight into the income generated from selling a certain number of products. In our problem, the revenue generated by selling toys is given by:\[ R = 500x - \frac{x^2}{25} \]Here, \( R \) is the revenue, and \( x \) is the number of toys sold. The term \( 500x \) represents the linear increase in revenue as more toys are sold, with each toy contributing \(500 to the total revenue. However, the term \(-\frac{x^2}{25}\) introduces a diminishing return, meaning that as more toys are produced and sold, there's a negative impact on revenue, possibly due to market saturation or increased competition.To determine how revenue changes as production changes, we differentiate the revenue function. The derivative, \( R'(x) = 500 - \frac{2x}{25} \), tells us how revenue changes with each additional toy. When substituting \( x = 5000 \), this means revenue decreases by \)150 per additional toy, indicating a diminishing return effect at this production level.

Profit Rate of Change

Profit is the net income, which is the difference between total revenue and total costs. To evaluate the change in profit, we use the derived rates of change for both revenue and costs. Profit is expressed as:\[ P = R - C \]The rate of change of profit is thus:\[ P'(x) = R'(x) - C'(x) \]From earlier steps, we found \( R'(x) = -150 \) and \( C'(x) = 1.05 \). Plugging these into the profit rate equation gives:\[ P'(x) = -150 - 1.05 = -151.05 \]This negative value indicates that profit is decreasing by $151.05 per additional toy produced. This highlights the importance of understanding the balance between production levels, cost management, and optimal pricing strategies to maximize profit. Understanding these metrics helps businesses make informed decisions about scaling production.