Let x, y be the volumes; m, n be the masses of two metallic cubes P and Q respectively. Each side of Q is two times that of P and mass of Q is two times that of P. Let u = m/x and v = n/y. Which one of the following is correct?

Correct Answer: u = 4v

Step-by-Step Explanation:

Let us analyze the given parameters for the two metallic cubes, P and Q, step-by-step.

1. Define the dimensions and volumes of cubes P and Q:

Let the side length of cube P be $s$.

The volume of a cube is given by the formula $V={\mathrm{side}}^3$.

Thus, the volume of cube P is:

$x=s^3$

According to the question, each side of cube Q is two times that of P. Therefore, the side length of cube Q is $2s$.

The volume of cube Q, denoted as $y$, is:

$y={\left(2s\right)}^3=8s^3=8x$

2. Define the masses of cubes P and Q:

Let the mass of cube P be $m$.

The mass of cube Q, denoted as $n$, is given to be two times that of P:

$n=2m$

3. Calculate the densities/ratios u and v:

We are given the expressions for $u$ and $v$ as follows:

$u=\frac{m}{x}$

$v=\frac{n}{y}$

Now, substitute $n=2m$ and $y=8x$ into the expression for $v$:

$v=\frac{2m}{8x}=\frac{1}{4}\cdot \frac{m}{x}$

Since $u=\frac{m}{x}$, we can replace $\frac{m}{x}$ with $u$:

$v=\frac{1}{4}u$

Multiplying both sides by 4 gives:

$u=4v$

Hence, the correct relation is u = 4v.