If the percentage increase in the radius of a cylinder is $300\%$, then by what percentage does the volume of the cylinder change, keeping the height of the cylinder constant?

Let the initial radius of the cylinder be $r$ and the constant height be $h$.
The original volume ($V_1$) is given by the formula:

$$V_1=\pi r^{2}h$$

The radius is increased by $300\%$, which means the new radius ($r^{\prime }$) becomes:

$$r^{\prime }=r+\left(\frac{300}{100}\times r\right)=r+3r=4r$$

Now, calculate the new volume ($V_2$) using the new radius $r^{\prime }$:

$$V_2=\pi (r^{\prime }{)}^{2}h=\pi (4r{)}^{2}h=16\pi r^{2}h=16V_1$$

The percentage change in the volume of the cylinder is:

$$\text{Percentage Change}=\left(\frac{V_2-V_1}{V_1}\right)\times 100\%$$

$$\text{Percentage Change}=\left(\frac{16V_1-V_1}{V_1}\right)\times 100\%=15\times 100\%=1500\%$$