Reference no: EM133375709

Mathematical Statistics

Problem 1. You and your friend wish to enter the brewing industry. In particular, you decide to distribute English ale in aluminum cans (cylinders sealed at both ends, naturally) that each have a volume of 355 cm3. As per usual, you want to minimize the total surface area of each can so as to save on material. Your friend claims that the ideal can has a height h that is twice its radius r. Is she right or wrong? Use Lagrange multipliers to check her claim, showing all of your work. (In the process, you will have to refa- miliarize yourself with the surface area and volume of a cylinder.) Report to three decimal places the dimensions (h and r) of the ideal can. As part of your answer, verify that these dimensions do indeed produce a minimal surface area. For this last step, use WolframAlpha. In your submission, please include screenshots of your WolframAlpha output.

Problem 2 (a). Make a sketch of the 1-form ω = (|sin(x)| + 1) dx supported culus is required for the calculation.)

(b). If ψ is a 5-form on R¹⁰, then ψ can be written as a linear combination of how many basic forms? (Give your answer as an integer.)
Now consider ψ ∧ ψ. Since it is a 10-form, recall that it can be written as

ψ ∧ ψ = F dx₁ ∧ dx₂ ∧ ..... ∧ dx₁₀ for some function F : R¹⁰ → R. Can we say what the function F must be? Explain very briefly.

(c). If θ is a 6-form on R¹⁰, then the volume of θ ∧ θ is 0. Is this true or false? Why? (Hint : θ ∧ θ is a k-form for which k?)

Problem 3. Use Fubini's Theorem to find the volume of ω = e_x1 cos(x₂) dx₁ ∧ dx₂ supported on

U = { (x₁, x₂ ) | 0 ≤ x₁ ≤ 1, Π/2 ≤ x₂ ≤ Π } ⊂ R².

What kind of volume is this (e.g. 1-volume, 2-volume, 3-volume)?