## find the volume of a solid from the surface areas of faces

Geometric formulae, word problems, theorems and proofs, etc.
little_dragon
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### find the volume of a solid from the surface areas of faces

The front, side, and bottom faces of a rectangular solid have areas of 24 square cm, 8 square cm, and 3 square cm, respectively. What is the volume of the solid, in cubic centimeters?

I drew a picture and labelled the sides as x, y, and z for the lengths. This makes the areas equal to xy = 24, xz = 8, and yz = 3. But what should I do next?

WishingBunny
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### Re: find the volume of a solid from the surface areas of faces

That's a good start. All you have to do now is find the express the value of two variables in terms of one common one.

I chose to solve that problem by expressing the value of x in terms of y, and the value of z in terms of y.

The equation that has x and y in it is xy = 24. I divided both sides of that equation by y to get x on one side, so the equation now states:
x = 24/y

The equation that has y and z in it is yz = 3. I divided both sides of that equation by y to get z on one side, so the equation now states:
z = 3/y

Then the third equation you came up with, xz = 8, comes into play. x = 24/y and z = 3/y, so I can substitute 24/y for x in the equation: xz = 8. I can also substitute 3/y for z in that same equation. So, xz = 8 becomes (24/y)(3/y) = 8.

Then you multiply the numerators and the denominators so your equation then reads, 72 over the y squared = 8.

You want to find the value of y so you multiply both sides of that equation by y squared. Then you get 72 = 8 times y squared.
Next, you divide both sides of the equation by 8, and so you get 9 = y squared. Then you find the square root of each side, leaving you with y = 3.

With that said, you can plug in 3 for the y in the other two equations, yz = 3 and xy = 24, to get the values of z and x.

If you do that, then you'll find that z = 1 and x = 8. Hope that helps

stapel_eliz
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This makes the areas equal to xy = 24, xz = 8, and yz = 3. But what should I do next?
Another method: If xy = 24 and xz = 8, then what is the value of (xy)/(xz)? How does this compare with the value of yz? What do you get if you take y/z = yz and divide through by y?

little_dragon
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### Re: find the volume of a solid from the surface areas of faces

Another method: If xy = 24 and xz = 8, then what is the value of (xy)/(xz)? How does this compare with the value of yz? What do you get if you take y/z = yz and divide through by y?
(xy)/(xz)=24/8=3=yz=y/z

divide off the y: z=1/z so z^2=1 so z=1 (because z=-1 doesn't work).

yz=3, so y=3. then x=8.

thanks!