A diagonal of one cube is 2 cm. A diagonal of another cube is 4*sqrt3 cm. The larger cube has volume 64 cubic cm. Find the volume of the smaller cube.
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There was this theorem mentioned in the lesson where if the scale factor of two similar solids is a to b, then
1) the ratio of corresponding perimeters is a to b.
2)the ratio of the base areas, of the lateral areas, and of the total areas is a squared to b squared.
3) the ratio of the volumes is a cubed to b cubed.
Does anybody know how to do this? Thanks for all of your help!!
Wnru
Yes, I can help you with that! According to the given information, we have two cubes with their respective diagonals. Let's call the side length of the smaller cube "a" and the side length of the larger cube "b".
We are given that the diagonal of the smaller cube is 2 cm. Since the diagonal of a cube forms a right triangle with two sides of equal length, we can use the Pythagorean theorem to find the length of one side. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (diagonal in this case) is equal to the sum of the squares of the other two sides.
So for the smaller cube, applying the Pythagorean theorem, we have:
a^2 + a^2 = 2^2
2a^2 = 4
a^2 = 2
a = √2 cm
Now, let's move on to the larger cube. We are given that the diagonal is 4√3 cm. Again, applying the Pythagorean theorem, we get:
b^2 + b^2 = (4√3)^2
2b^2 = 48
b^2 = 24
b = √24 cm
To find the volume of the smaller cube, we can use the third property of similar solids that you mentioned earlier. According to this property, the ratio of the volumes of similar solids is equal to the cube of the ratio of their corresponding side lengths.
So, the ratio of the side lengths is a/b = (√2)/ (√24).
To simplify this ratio, we can rationalize the denominator by multiplying both the numerator and denominator by √24. This gives us:
(√2 * √24) / (√24 * √24) = (√(2*24)) / 24 = √48 / 24 = (√16 * √3) / 24 = (4 * √3) / 24 = √3 / 6
Therefore, the ratio of the side lengths is √3 / 6.
Now we can use the given information that the volume of the larger cube is 64 cubic cm to find the volume of the smaller cube.
Using the property we discussed earlier, the ratio of the volumes is equal to the cube of the ratio of the side lengths. So:
(√3 / 6)^3 = (√3 / 6) * (√3 / 6) * (√3 / 6)
= (√3 * √3 * √3) / (6 * 6 * 6)
= 3√3 / 216
Since the volume of the larger cube is 64 cubic cm, we can set up the following proportion:
(3√3 / 216) : 64 = (√2) : V
where V represents the volume of the smaller cube.
To solve for V, we can cross multiply and then solve for V:
(3√3 / 216) * V = (√2) * 64
Simplifying the equation:
V = (√2 * 64 * 216) / (3√3)
V = (2√2 * 216) / (3√3)
Finally, we can simplify the expression further:
V = (432√2) / (3√3)
V = 144√2 / √3
V ≈ 144.47 cubic cm
So, the volume of the smaller cube is approximately 144.47 cubic cm.
The theorem you mentioned is true but I won't try to prove it here. In your case, since a linear dimension of the cube increases by a factor of 2 sqrt 3 compared to the smaller cube, The larger cube has a volume that is (2 sqrt3)^3 larger.
That equals 2^3 * 3^(3/2)= 41.57 times
(or 24*sqrt3) larger in volume