Anonymous
Anonymous asked in 科學及數學數學 · 8 years ago

F.2 MATH. AREA

claudia is designing a cylindrical can that will hold 525mL of soup. She uses the design that minimize the surface area. What is the radius of the can?

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  • 8 years ago
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    I use algebra method instead of differentiation since this is a F.2 Maths question.

    Assuming the can have a cover.

    Note that 1cm³ = 1mL.

    Let r (cm) be the radius and h (cm) be the height of the can , then

    the volume

    = πr² h = 525

    the total surface area

    = 2πr² + 2πrh

    = 2πr² + πrh + πrh

    ≥ 3 ∛( 2πr² * πrh * πrh).. A.M.≥ G.M. : (a + b + c)/3 ≥ ∛(a b c) for a ,b ,c ≥ 0.

    = 3 ∛( 2π (πr² h)² )

    = 3 ∛( 2π (525)² ) ,

    "=" holds if and only if 2πr² = πrh = πrh , i.e. 2r = h , substitute into πr² h = 525 :πr² 2r = 525

    r³ = 525 / (2π)

    r = ∛[525 / (2π)] = 4.37...

    The radius of the can = 4.37... cm

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