### Math Notes

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#### Solid Geometry Solutions

##### Topics || Problems

If a cube has an edge equal to the diagonal of another cube, what is the ratio of their volumes?

Let $$a$$ be the side of the first cube and $$b$$ be the diagonal of the second cube and $$v_a$$ and $$v_b$$ be their volumes respectively.

$$v_a = a^3$$

If x is the side of the second cube then,

$$b^2 = (\sqrt{2} x)^2 +x^2$$ $$b^2 = 3x^2$$
$$x = \frac{b}{\sqrt{3}}$$
$$v_b = x^3$$
$$v_b = (\frac{b}{\sqrt{3}})^3$$
$$v_b = \frac{b^3}{3^{\frac{3}{2}}}$$

Thus the ratio between the volume of the two cubes is,

$$\frac{v_a}{v_b} = \frac{a^3}{\frac{b^3}{3^{\frac{3}{2}}}}$$

$$\frac{v_a}{v_b} = \frac{a^3}{\frac{a^3}{3^{\frac{3}{2}}}}$$ , but $$a = b$$

$$\frac{v_a}{v_b} = \frac{1}{\frac{1}{3^{\frac{3}{2}}}}$$

$$\frac{v_a}{v_b} = 3^{\frac{3}{2}}$$