Related Rate Problem #5: Cobbs-Douglas Production Function

Sun Surfer Math
1 Apr 202209:13
EducationalLearning
32 Likes 10 Comments

TLDRThe video script presents a related rate problem involving a Cobbs Douglas production function. The company aims to maintain a constant workforce of 129 workers, with a current demand of 997 mass spectrometers per year, increasing by 87 spectrometers annually. The challenge is to find the rate at which the daily operating budget (y) should increase to meet this growing demand. The production function given is p = 9 * x^0.3 * y^0.7, where p is the demand, x is the number of workers, and y is the daily operating budget. By applying the product rule and substituting known values, the script derives the rate of change of y with respect to time (dy/dt). After solving for y using the given p and x values, the script calculates dy/dt to be 14.74, indicating the rate at which the daily operating budget must increase to accommodate the rising demand.

Takeaways
  • ๐Ÿ“ˆ The problem involves a related rate using the Cobbs-Douglas production function, focusing on how to increase the daily operating budget to meet production demands.
  • ๐Ÿ‘ท The company maintains a constant workforce of 129 workers, which means the change in employees with respect to time (dx/dt) is zero.
  • ๐Ÿ“‰ The current demand is 997 mass spectrometers per year, and it is increasing by 87 spectrometers per year (dp/dt = 87).
  • ๐Ÿ’ก The goal is to find the rate at which the daily operating budget (y) should increase with respect to time (dy/dt).
  • ๐Ÿ” The production function is given by p = 9 * (x^0.3) * (y^0.7), where p is the demand, x is the number of workers, and y is the daily operating budget.
  • ๐Ÿงฎ To solve for dy/dt, the derivative of the production function is taken using the product rule, resulting in a complex expression involving both x and y.
  • ๐Ÿ”‘ The derivative expression is simplified by combining terms and isolating dy/dt, which is the rate we are trying to find.
  • โœ… Plugging in the known values (dp/dt = 87, x = 129) and solving for y, we find that y equals 103.7.
  • ๐Ÿ“Œ Substituting the values of x, y, and dp/dt into the derivative expression, we can calculate the value of dy/dt.
  • ๐Ÿง The second part of the derivative expression involving dx/dt is zero since the workforce is constant, simplifying the calculation.
  • ๐Ÿ“ After performing the algebra and arithmetic, the calculated rate of increase for the daily operating budget with respect to time (dy/dt) is 14.74.
Q & A
  • What is the Cobb-Douglas production function used in this problem?

    -The Cobb-Douglas production function used in this problem is represented by the equation P = 9 * X^0.3 * Y^0.7, where P is the production output, X is the number of employees, and Y is the daily operating budget.

  • Why is the rate of change of employees (dx/dt) equal to zero in this problem?

    -The rate of change of employees (dx/dt) is zero because the company plans to maintain a constant workforce of 129 workers. Therefore, there is no change in the number of employees over time.

  • What does dp/dt represent in this scenario, and what is its value?

    -In this scenario, dp/dt represents the rate of change of production output with respect to time. Its value is 87, indicating that the production of mass spectrometers is increasing by 87 units per year.

  • How is the value of Y determined in the video?

    -The value of Y, the daily operating budget, is determined by solving the Cobb-Douglas production function P = 9 * X^0.3 * Y^0.7 with known values of P (997 spectrometers per year) and X (129 employees), which yields Y as approximately 103.7.

  • What is the primary goal of the calculation shown in the video?

    -The primary goal of the calculation shown in the video is to determine how fast the daily operating budget (Y) should be increasing, quantified as dy/dt, in order to meet the increasing demand for production.

  • Why is the second part of the derivative set to zero?

    -The second part of the derivative is set to zero because dx/dt equals zero, due to the constant number of employees. This results in the term involving dx/dt in the derivative equation contributing nothing to the rate of change.

  • What does the term 6.3 * (X/Y)^0.3 represent in the derivative calculation?

    -The term 6.3 * (X/Y)^0.3 in the derivative calculation represents the part of the derivative involving the rate of change of the daily operating budget (dy/dt), scaled by the ratio of employees to the operating budget raised to the power of 0.3, and multiplied by 6.3.

  • How is the derivative of the Cobb-Douglas production function calculated?

    -The derivative of the Cobb-Douglas production function is calculated using the product rule, applied to the function parts involving X and Y, given that both are treated as dependent variables in this context.

  • What is the final value of dy/dt and its significance?

    -The final calculated value of dy/dt is 14.74, which indicates the rate at which the daily operating budget should increase per unit time to meet the increasing production demands.

  • What mathematical techniques are primarily used in solving the related rates problem in this video?

    -The primary mathematical techniques used include the product rule for differentiation and algebraic manipulation to isolate and solve for the variable dy/dt.

Outlines
00:00
๐Ÿ“ˆ Understanding the Cobb-Douglas Production Function Problem

This paragraph introduces a related rate problem involving the Cobb-Douglas production function. The video aims to identify known variables: a constant workforce of 129 workers (x=129), which implies no change in the number of employees over time (dx/dt=0), and a current demand of 997 mass spectrometers per year (p=997) increasing by 87 per year (dp/dt=87). The goal is to find the rate at which the daily operating budget (y) should increase (dy/dt). The production function given is p = 9x^0.3y^0.7. To solve for dy/dt, the derivative of the production function is taken using the product rule, resulting in an equation involving dx/dt and dy/dt. Since dx/dt is zero (due to the constant workforce), the equation simplifies to focus on dy/dt.

05:00
๐Ÿงฎ Solving for the Rate of Change in the Operating Budget

The second paragraph is focused on solving for the rate of change in the operating budget (dy/dt) using algebraic manipulation. The numbers for dp/dt, x, and y are plugged into the derivative equation. The value of y is found by rearranging the original production function equation with the known values of p and x, which gives y=103.7. Substituting all known values into the derivative equation and simplifying, the rate of increase in the operating budget (dy/dt) is calculated to be 14.74. This final value represents how fast the daily operating budget should increase to meet the increasing demand for mass spectrometers.

Mindmap
Keywords
๐Ÿ’กRelated Rate
Related Rate problems involve finding the rate at which one quantity changes with respect to another, when both are changing simultaneously. In the video, the related rate problem is approached using a production function to determine how fast the daily operating budget should increase to meet the growing demand for mass spectrometers.
๐Ÿ’กCobbs Douglas Production Function
The Cobbs Douglas Production Function is a specific type of production function that relates output (product quantity) to inputs (factors of production). It is often used in economics to model the production process. In the video, this function is used to express the relationship between the number of workers (x), the daily operating budget (y), and the production output (p).
๐Ÿ’กConstant Workforce
A constant workforce implies that the number of employees remains unchanged over time. In the context of the video, it is given that the company plans to maintain a constant workforce of 129 workers, which means the rate of change of employees with respect to time (dx/dt) is zero.
๐Ÿ’กMass Spectrometers
Mass spectrometers are analytical instruments that are used for determining the mass and chemical structure of molecules. In the video, the company's production is focused on these instruments, with a current demand of 997 units per year that is increasing.
๐Ÿ’กDerivative
In calculus, a derivative represents the rate at which a function changes with respect to a variable. The video involves taking the derivative of the production function to find the rate of change of the daily operating budget (dy/dt) with respect to time.
๐Ÿ’กProduct Rule
The product rule is a fundamental theorem in calculus used to find the derivative of a product of two functions. In the video, the product rule is applied to differentiate the production function with respect to time, which involves two variables, x and y.
๐Ÿ’กDaily Operating Budget
The daily operating budget refers to the amount of money allocated for a company's day-to-day operations. In the video, the goal is to find out how quickly this budget should increase (dy/dt) to meet the rising demand for mass spectrometers.
๐Ÿ’กAlgebra
Algebra is a branch of mathematics that uses symbols and rules to manipulate and solve equations. In the context of the video, algebra is used to solve for the unknown variable y in the production function and subsequently to find the rate of change of y with respect to time (dy/dt).
๐Ÿ’กRate of Change
The rate of change, often denoted as dy/dt or dx/dt, is a mathematical expression that describes how one quantity changes in relation to another. In the video, the rate of change of the demand for mass spectrometers (dp/dt) is given, and the task is to find the rate of change of the daily operating budget (dy/dt).
๐Ÿ’กProduction
Production refers to the process of creating goods or services from inputs such as labor, capital, and raw materials. The video's theme revolves around increasing production to meet a growing demand, which is mathematically modeled using the Cobbs Douglas Production Function.
๐Ÿ’กSolving for dy/dt
Solving for dy/dt involves finding the rate at which the daily operating budget y changes with respect to time. This is the central problem in the video, where the derivative is manipulated and solved using the given information and the production function.
Highlights

The video discusses a related rate problem using the Cobbs Douglas production function.

The company maintains a constant workforce of 129 workers, implying no change in the number of employees over time.

The current demand is 997 mass spectrometers per year, which is increasing by 87 spectrometers per year.

The goal is to find the rate at which the daily operating budget (y) should increase to meet the increasing demand.

The production function is given by p = 9 * x^0.3 * y^0.7, where p is the demand, x is the number of workers, and y is the daily operating budget.

The derivative of the production function is taken using the product rule due to the presence of two independent variables, x and y.

The derivative is simplified to express the relationship between the rates of change of p, x, and y.

The problem involves algebraic manipulation to isolate the rate of change of y with respect to time (dy/dt).

The value of y is found by substituting known values into the original production function equation.

The value of y is calculated to be 103.7, which is crucial for further calculations.

All known values are plugged into the derivative to solve for dy/dt, with dx/dt being zero due to the constant workforce.

The final calculation for dy/dt results in a value of 14.74, indicating the rate at which the daily operating budget should increase.

The video provides a step-by-step guide on solving related rate problems in the context of production functions.

The problem-solving approach emphasizes the importance of identifying constants and rates of change in economic models.

The use of the Cobbs Douglas production function in the problem highlights its applicability in real-world economic scenarios.

The video demonstrates the practical application of calculus in economic decision-making processes.

Attention to detail is crucial, as shown by the careful consideration of the constant workforce and its implications on the problem.

The video concludes with a clear solution to the related rate problem, showcasing the effectiveness of the chosen method.

Transcripts
Rate This

5.0 / 5 (0 votes)

Thanks for rating: