The figure shown is a right circular cylinder with a radius of r and height of

Choice C is correct. The volume of a right circular cylinder is equal to $\pi a^{2}b$, where $a$ is the radius of a base of the cylinder and $b$ is the height of the cylinder. It’s given that the cylinder shown has a radius of $r$ and a height of $h$. It follows that the volume of the cylinder shown is equal to $\pi r^{2}h$. It’s given that the second right circular cylinder has a radius of $R$ and a height of $H$. It follows that the volume of the second cylinder is equal to $\pi R^{2}H$. Choice C gives $R=7r$ and $H=8h$. Substituting $7r$ for $R$ and $8h$ for $H$ in the expression that represents the volume of the second cylinder yields $\pi {\left(7r\right)}^2\left(8h\right)$, or $\pi \left(49r^2\right)\left(8h\right)$, which is equivalent to $\pi \left(392r^{2}h\right)$, or $392\left(\pi r^{2}h\right)$. This expression is equal to $392$ times the volume of the cylinder shown, $\pi r^{2}h$. Therefore, $R=7r$ and $H=8h$ could represent the radius $R$, in terms of $r$, and the height $H$, in terms of $h$, of the second cylinder.

Choice A is incorrect. Substituting $8r$ for $R$ and $7h$ for $H$ in the expression that represents the volume of the second cylinder yields $\pi {\left(8r\right)}^2\left(7h\right)$, or $\pi \left(64r^2\right)\left(7h\right)$, which is equivalent to $\pi \left(448r^{2}h\right)$, or $448\left(\pi r^{2}h\right)$. This expression is equal to $448$, not $392$, times the volume of the cylinder shown. 

Choice B is incorrect. Substituting $8r$ for $R$ and $49h$ for $H$ in the expression that represents the volume of the second cylinder yields $\pi {\left(8r\right)}^2\left(49h\right)$, or $\pi \left(64r^2\right)\left(49h\right)$, which is equivalent to $\pi \left(\mathrm{3,136}{r}^{2}h\right)$, or $\mathrm{3,136}\left(\pi r^{2}h\right)$. This expression is equal to $\mathrm{3,136}$, not $392$, times the volume of the cylinder shown.

Choice D is incorrect. Substituting $49r$ for $R$ and $8h$ for $H$ in the expression that represents the volume of the second cylinder yields $\pi {\left(49r\right)}^2\left(8h\right)$, or $\pi \left(\mathrm{2,401}{r}^2\right)\left(8h\right)$, which is equivalent to $\pi \left(\mathrm{19,208}{r}^{2}h\right)$, or $\mathrm{19,208}\left(\pi r^{2}h\right)$. This expression is equal to $\mathrm{19,208}$, not $392$, times the volume of the cylinder shown.