The volume of $A$ in terms of $C$

Question

let $A$ be a region in $\mathbb R^3$ such that $$x_1^4+x_2^4+x_3^4\le 1 \text{ and } x_i\ge0$$

let $C$ be the volume of this region, now consider the same region but instead of $1$ we have $$x_1^4+x_2^4+x_3^4\le 29$$

what is the volume of this region in terms of $C$.

I was taught that if a region in $\mathbb R^3$ got multiplied by a factor of $r$ then the new volume is just $r^3 \cdot (\text{the original volume})$, so our volume is just $29^3C$, isn't it?

Answer 1

$$x_1^4+x_2^4+x_3^4\le 29 \implies \left(\frac{x_1}{\sqrt[4]{29}}\right)^4+\left(\frac{x_2}{\sqrt[4]{29}}\right)^4+\left(\frac{x_4}{\sqrt[4]{29}}\right)^4 \le 1$$

The volume is so scaled by a factor of $\sqrt[4]{29}$, in other word, the new volume is equal to $29^{\frac{3}{4}}C $.