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Math Help - volume of a cube

  1. #1
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    Question volume and surface areas of a cube and rectangular solids

    can you help me solve this problems ?

    im a little bit confused about them

    1. One cube has a face equivalent to the total area of another cube. find the ratio of their volume.

    2. Find the approximate change in the volume of a rectangular solid caused by increasing each side by 5%

    thanks
    Last edited by FailCalculus; October 11th 2009 at 01:32 AM.
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  2. #2
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    Quote Originally Posted by FailCalculus View Post
    can you help me solve this problems ?

    im a little bit confused about them

    1. One cube has a face equivalent to the total area of another cube. find the ratio of their volume.

    2. Find the approximate change in the volume of a rectangular solid caused by increasing each side by 5%

    thanks
    to #1: Let a denote the length of the edge of the smaller cube then the surface area of this cube is

    A_{small} = 6a^2 and it's volume is V_{small} = a^3

    Let b denote the length of the edge of the larger cube then the area of one face is
    <br />
b^2 = 6a^2~\implies~b=a\sqrt{6}

    Then the volume of the larger cube is

    V_{large} = b^3 = \left(a\sqrt{6}\right)^3 = 6a^3 \sqrt{6}

    Calculate the ration \dfrac{V_{large}}{V_{small}}

    to #2: Let a, b and c denote the edges of the solid. Then the volume is calculated by V = a\cdot b \cdot c
    If you increase the edge a of a solid by 5% then you get a + \frac5{100} a = 1.05 a

    Calculate the lenghtes of the 3 edges and determine the new volume. Compare this value with the initial volume.
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  3. #3
    Senior Member pacman's Avatar
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    1. One cube has a face equivalent to the total area of another cube. find the ratio of their volume.

    -----------------------------------------------------------------------------
    1. Let A = X^2 = surface area of the cube, X is the length of side,

    then X = sqrt of A

    Also a = x^2 = surface area of the new cube, x is the length of the new cube,

    x = sqrt of a

    but A = 6a, that is X^2 = 6x^2, cross-multiplying,

    (X/x)^2 = 6

    (X/x) = sqrt 6

    Volume ratio =

    (Vol. of Cube)/(Vol. of new cube) = (X/x)^3 = (sqrt 6)^3 = 14.7 cubic units
    -------------------------------------------------------------------------------------
    2)Find the approximate change in the volume of a rectangular solid caused by increasing each side by 5%

    % Increase = [New Volume - Old Volume](100%)/Old Volume

    = [LWH - lwh](100%)/lwh

    = [(1.05 l)(1.05 w)(1.05 h) - lwh](100%)/(lwh), cancel lwh

    % Increase = [(1.05)(1.05)(1.05) - 1](100)% = 0.0125% increase in volume
    Last edited by pacman; October 11th 2009 at 08:35 AM.
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  4. #4
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    @earboth and pacman
    thanks
    Last edited by mr fantastic; October 11th 2009 at 01:56 AM. Reason: Tagged on question moved to a new thread.
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  5. #5
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    Quote Originally Posted by pacman View Post
    ...
    2)Find the approximate change in the volume of a rectangular solid caused by increasing each side by 5%

    % Increase = [New Volume - Old Volume](100%)/Old Volume

    = [LWH - lwh](100%)/lwh

    = [(1.05 l)(1.05 w)(1.05 h) - lwh](100%)/(lwh), cancel lwh

    Volume Ratio = [(1.05)(1.05)(1.05) - 1](100)% = 0.0125% increase in volume
    I don't want to pick at you because all your considerations and calculations are Ok so far but I'm very interested how you got your result. It differs a lot from the one I got:

    ((1.05)^3-1) \cdot 100\% \approx 15.8\ \%
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  6. #6
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    Quote Originally Posted by pacman View Post
    1. One cube has a face equivalent to the total area of another cube. find the ratio of their volume.

    -----------------------------------------------------------------------------
    1. Let A = X^2 = surface area of the cube, X is the length of side,
    Clairification: this is the area of one face of a cube. That's why you multiply by 6 below.

    then X = sqrt of A

    Also a = x^2 = surface area of the new cube, x is the length of the new cube,

    x = sqrt of a

    but A = 6a, that is X^2 = 6x^2, cross-multiplying,

    (X/x)^2 = 6

    (X/x) = sqrt 6

    Volume ratio =

    (Vol. of Cube)/(Vol. of new cube) = (X/x)^3 = (sqrt 6)^3 = 14.7 cubic units
    -------------------------------------------------------------------------------------
    2)Find the approximate change in the volume of a rectangular solid caused by increasing each side by 5%

    % Increase = [New Volume - Old Volume](100%)/Old Volume

    = [LWH - lwh](100%)/lwh

    = [(1.05 l)(1.05 w)(1.05 h) - lwh](100%)/(lwh), cancel lwh

    Volume Ratio = [(1.05)(1.05)(1.05) - 1](100)% = 0.0125% increase in volume
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  7. #7
    Senior Member pacman's Avatar
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    You are right earboth, it should be % Increase = [(1.05)(1.05)(1.05) - 1](100)% = 15.7625%.




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  8. #8
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    now i get it . thanks for the help of both of you
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