Problem 57
Question
MANUFACTURING PROFIT The total revenue realized in the sale of \(x\) units of the LectroCopy photocopying machine is $$ -0.04 x^{2}+2000 x $$ dollars/week, and the total cost incurred in manufacturing \(x\) units of the machines is $$ 0.000002 x^{3}-0.02 x^{2}+1000 x+120,000 $$ dollars/week \((0 \leq x \leq 50,000)\). Find an expression giving the total weekly profit of the company.
Step-by-Step Solution
Verified Answer
The total weekly profit of the company, as a function of the number of machines (\(x\)), is
$$
P(x) = -0.000002x^3 - 0.02x^2 + 1000x - 120000
$$
1Step 1: Write down the expression for total revenue and cost
From the given exercise, we know that the expression for the total revenue in the sale of \(x\) machines is
$$
-0.04x^2 + 2000x
$$
and the expression for the total cost of manufacturing \(x\) machines is
$$
0.000002x^3 - 0.02x^2 + 1000x + 120000.
$$
2Step 2: Calculate the profit expression by subtracting the cost from revenue
To find the profit expression of the company, we will subtract the total cost expression from the total revenue expression. That is
$$
P(x) = (-0.04x^2 + 2000x) - (0.000002x^3 - 0.02x^2 + 1000x + 120000)
$$
3Step 3: Simplify the profit expression
Now let's simplify the expression for profit by combining like terms:
$$
P(x) = -0.04x^2 + 2000x - 0.000002x^3 + 0.02x^2 - 1000x - 120000
$$
Rearrange and combine the terms with the same powers of x:
$$
P(x) = (-0.000002x^3) + (0.02x^2 - 0.04x^2) + (2000x - 1000x) - 120000
$$
$$
P(x) = -0.000002x^3 - 0.02x^2 + 1000x - 120000
$$
4Step 4: Write down the final expression for profit
The expression for the total weekly profit of the company, as a function of the number of machines (\(x\)), is
$$
P(x) = -0.000002x^3 - 0.02x^2 + 1000x - 120000
$$
Key Concepts
Revenue and Cost FunctionsQuadratic and Cubic FunctionsProfit Calculation
Revenue and Cost Functions
Understanding the revenue and cost functions in business contexts is crucial for assessing financial performance. Revenue functions represent the money a company receives from selling its products, typically dependent on the quantity sold. For instance, if a photocopying machine is sold for a certain price, the revenue function would mathematically model the total income from selling 'x' number of machines.
Cost functions, on the other hand, detail the total expenses incurred in manufacturing goods. These encompass direct costs such as materials, labor, and also indirect costs like overhead. In our example, the cost function for producing 'x' photocopying machines includes both a cubic term to represent growing costs at different production levels and fixed costs, capturing consistent expenses regardless of the quantity produced.
Both functions come alive in a profit-driven company. By representing them mathematically, businesses can manipulate these functions to predict outcomes, optimize production, and adjust pricing strategies to ensure sustainability and growth.
Cost functions, on the other hand, detail the total expenses incurred in manufacturing goods. These encompass direct costs such as materials, labor, and also indirect costs like overhead. In our example, the cost function for producing 'x' photocopying machines includes both a cubic term to represent growing costs at different production levels and fixed costs, capturing consistent expenses regardless of the quantity produced.
Both functions come alive in a profit-driven company. By representing them mathematically, businesses can manipulate these functions to predict outcomes, optimize production, and adjust pricing strategies to ensure sustainability and growth.
Quadratic and Cubic Functions
Quadratic and cubic functions are types of polynomial functions with distinct characteristics and graphs. A quadratic function, which appears as \( ax^2 + bx + c \), creates a parabola when graphed. This U-shaped curve can represent various dynamics but is notably applied in revenue and cost analysis to depict diminishing returns or increased costs after a certain production level.
Cubic functions, formed as \( ax^3 + bx^2 + cx + d \), present more complexity with the possibility of having one or two turning points, implying a more dynamic relationship between the units produced and the total cost. In our manufacturing profit scenario, the presence of a cubic term in the cost function may indicate that as production ramps up, the costs do not simply increase at a steady rate but can accelerate, reflecting the increased complexity and resources needed at higher production levels.
Graph analysis of these functions helps businesses understand where they might achieve maximum productivity and efficiency, leading to greater profitability.
Cubic functions, formed as \( ax^3 + bx^2 + cx + d \), present more complexity with the possibility of having one or two turning points, implying a more dynamic relationship between the units produced and the total cost. In our manufacturing profit scenario, the presence of a cubic term in the cost function may indicate that as production ramps up, the costs do not simply increase at a steady rate but can accelerate, reflecting the increased complexity and resources needed at higher production levels.
Graph analysis of these functions helps businesses understand where they might achieve maximum productivity and efficiency, leading to greater profitability.
Profit Calculation
Profit calculation is the act of determining the financial gain a company makes after all expenses are paid. In mathematical terms, profit is calculated by subtracting the total costs from the total revenue. The resulting expression is known as the profit function \( P(x) \), reflecting the profit based on different levels of production or sales.
In our manufacturing profit example, the profit function is derived by subtracting the cost function from the revenue function. Simplifying this yields a profit function that combines both quadratic and cubic elements. Analyzing this function gives insights into how many units a company needs to sell to break even or achieve desired profit margins. Factors like the number of products sold, production costs, market conditions, and sales price all feed into this profit function.
Understanding and optimizing the profit function can lead companies to make informed choices about production levels, cost-cutting, and pricing strategies—key components that drive fiscal health and ensure an organization's long-term success.
In our manufacturing profit example, the profit function is derived by subtracting the cost function from the revenue function. Simplifying this yields a profit function that combines both quadratic and cubic elements. Analyzing this function gives insights into how many units a company needs to sell to break even or achieve desired profit margins. Factors like the number of products sold, production costs, market conditions, and sales price all feed into this profit function.
Understanding and optimizing the profit function can lead companies to make informed choices about production levels, cost-cutting, and pricing strategies—key components that drive fiscal health and ensure an organization's long-term success.
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