At www.mathsassignmenthelp.com, we take pride in helping students excel in mathematics by providing personalized and expert guidance. If you’re looking for help with Math Assignment, you’ve come to the right place. Today, we present a glimpse into how our experts solve advanced mathematical questions. These questions are typical of master-level courses, showcasing the depth and precision we bring to our work.

Question 1: Optimization in Real-World Problems

A manufacturing company wants to minimize the cost of producing a cylindrical storage tank that must hold a fixed volume of 1000 cubic meters. The tank has a top and bottom, which are circular, and the material for the curved surface costs twice as much per square meter as the material for the top and bottom. What are the dimensions of the tank that will minimize the cost?

Solution:

Optimization problems like this require us to minimize the cost function while meeting the volume constraint. Let’s break it down step by step:

  1. Define Variables:

    • Let the radius of the base be meters.

    • Let the height of the cylinder be meters.

  2. Volume Constraint:

    • The volume of a cylinder is given by:

    • Substituting the required volume:

  3. Cost Function:

    • The cost involves the curved surface area and the two circular ends.

    • Surface area of the curved part:

    • Surface area of the top and bottom:

    • Since the curved surface is twice as expensive, the total cost is:

    • Substituting :

  4. Minimize Cost Function:

    • Differentiate with respect to and set it to zero: Approximation gives meters.

    • Find : Approximation gives meters.

Thus, the tank dimensions that minimize cost are approximately meters and meters.

Question 2: Eigenvalues and Their Applications

Find the eigenvalues and eigenvectors of the matrix:

Solution:

Eigenvalues and eigenvectors are foundational in advanced linear algebra and have numerous applications in fields like data analysis, physics, and engineering.

  1. Definition:

    • Eigenvalues satisfy the equation: where is the given matrix, is the identity matrix, and is the eigenvector.

  2. Find the Eigenvalues:

    • Start with the determinant:

    • Expand:

    • Factorize:

  3. Find Eigenvectors:

    • For : Solve : Simplifying gives . Let .

    • For : Solve : Simplifying gives . Let .

Thus, the eigenvalues are and , with corresponding eigenvectors and .

These examples demonstrate the analytical rigor and precision our experts at www.mathsassignmenthelp.com provide. If you need help with Math Assignment, we are here to guide you every step of the way. Feel free to reach out to us for comprehensive solutions tailored to your academic needs.

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