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Math Help - Changing dimension in a rectangle

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    Changing dimension in a rectangle

    so where do i start for this problem?

    the length l of a rectangle is decreasing at the rate of 2 cm/sec while the width w is increasing at the rate of 2 cm/sec. when l=12cm and w= 5cm, find the rates of change of area, perimeter, and the lengths of the diagonals of the rectangle.
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  2. #2
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    Quote Originally Posted by cyberdx16 View Post
    so where do i start for this problem?

    the length l of a rectangle is decreasing at the rate of 2 cm/sec while the width w is increasing at the rate of 2 cm/sec. when l=12cm and w= 5cm, find the rates of change of area, perimeter, and the lengths of the diagonals of the rectangle.
    First we know that,
    A=w*l

    Now A,w,l are function of t (time)
    Thus, (differenciating and using product rule),
    dA/dt=w*dl/dt+l*dw/dt
    We are given that,
    dl/dt=-2 cm/sec
    dw/dt=2 cm/sec
    And, w=5 cm with l=12 cm
    Thus,
    dA/dt=(5)(-2)+12(2)=10 cm^2/sec

    For perimeter consider,
    P=2w+2l
    Thus,
    dP/dl=2(dw/dt)+2(dl/dt)
    Thus,
    dP/dl=2(2)+2(-2)=0 cm/sec
    Which makes sense since if you are increasing one and decreasing one at the same rate there is no change in rate.

    For the diagnol, (by Pythagorean theorem)
    d^2=l^2+w^2
    Thus, (implicit differenciation)
    2d*(dd/dt)=2l(dl/dt)+2w(dw/dt)
    When (l,w)=(5,12) we have d=13
    2(13)(dd/dt)=2(5)(-2)+2(12)(2)
    Thus,
    26(dd/dt)=28
    Thus,
    dd/dt=28/26 cm/sec
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