By what percentage will the transmission range of a TV tower be affected when the height of the tower is increased by 21%?

The transmission range of a TV tower is given by:

$$d = \sqrt{2Rh}$$

where $$R$$ is the radius of the Earth and $$h$$ is the height of the tower. So $$d \propto \sqrt{h}$$.

When the height is increased by 21%, the new height is $$h' = 1.21h$$. Now the new range becomes:

$$d' = \sqrt{2R \cdot 1.21h} = \sqrt{1.21} \cdot \sqrt{2Rh} = 1.1 \cdot d$$

The percentage increase in range is:

$$\frac{d' - d}{d} \times 100 = (1.1 - 1) \times 100 = 10\%$$

Hence, the transmission range is increased by $$10\%$$.