# Thread: volume of a cube

1. ## 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

2. Originally Posted by FailCalculus
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

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

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

Then the volume of the larger cube is

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

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

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

Calculate the lenghtes of the 3 edges and determine the new volume. Compare this value with the initial volume.

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

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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
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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

4. @earboth and pacman
thanks

5. Originally Posted by pacman
...
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:

$\displaystyle ((1.05)^3-1) \cdot 100\% \approx 15.8\ \%$

6. Originally Posted by pacman
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

7. You are right earboth, it should be % Increase = [(1.05)(1.05)(1.05) - 1](100)% = 15.7625%.

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

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# one cube has a face equivalent to the total area of another cube.Find the ratio of their volumes.

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