Math

how could you find the measurements of the frame with a 16 inch perimeter antd the greatest possible area?

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  1. A square of 4 by 4 would give you the largest area

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  2. P=Perimeter

    A=Area

    a,b=sides of rectacangle

    P=2a+2b

    P=2(a+b) Divide with 2

    P/2=a+b

    a+b=P/2

    a+b=16/2

    a+b=8

    b=8-a

    A=a*b

    A=a*(8-a)

    A=8a-a^2

    A=-a^2+8a

    Function have extreme value in point where first derivative=0

    First derivative:

    dA/da=-2a+8

    dA/da=0

    -2a+8=0

    8=2a Divide with 2

    a=4in

    b=8-a

    b=8-4

    b=4in

    Second derivative:

    d^2A/da^2=-2

    When second derivative is negative tion have maximum.

    a=4in b=4 in

    A(max)=4*4

    A(max)=16in^2

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