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9 July, 08:12

a circular window with a circumference of 10 feet is to be placed into a wall at a new home. The builder cuts a hole for the window and says that it has an area of at least 8 square feet. explain why she is right or wrong. Show work.

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  1. 9 July, 09:38
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    Builder is right.

    Step-by-step explanation:

    circumference, C = 10 feet

    Let the radius of circular window is r.

    Use the formula for circumference.

    C = 2 x π x r = 10 ... (1)

    10 = 2 x π x r

    r = 1.59 ft

    The formula for the area of circular window

    A = π r²

    Differentiate with respect to r.

    dA/dr = 2 x π x r

    Differentiate again with respect to r.

    d²A/dr² = 2 π

    It is positive for any value of r so the area of minimum.

    So, minimum area is

    A = 3.14 x 1.59 x 1.59 = 8 ft

    So, the minimum area id 8 ft². The builder is right.
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