To double the covering range of a TV transmitting tower, its height should be multiplied by:

For a television transmitting tower the ground‐range up to which its signals can reach by line-of-sight propagation is governed by the well-known horizon formula

$$d=\sqrt{2\,R\,h},$$

where $$d$$ is the maximum coverage distance (range), $$h$$ is the height of the tower, and $$R$$ is the mean radius of the Earth. We shall treat $$R$$ as a constant because it is enormously larger than any practical tower height.

Let the original height of the tower be $$h$$; then its original coverage range is

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

Now the problem says that we want to double this range. Hence the new desired range, which we denote by $$d'$$, must satisfy

$$d'=2d.$$

Suppose we raise the tower to a new height $$h'$$. The horizon formula applied to this new height gives

$$d'=\sqrt{2R\,h'}.$$

We are therefore required to solve the equation

$$\sqrt{2R\,h'} = 2d.$$

Substituting the original value of $$d$$ in terms of $$h$$ on the right-hand side, we have

$$\sqrt{2R\,h'} = 2\bigl(\sqrt{2Rh}\bigr).$$

Squaring both sides eliminates the square roots:

$$2R\,h' = 4\,(2R\,h).$$

Simplifying the right-hand side gives

$$2R\,h' = 8R\,h.$$

We can now divide both sides by $$2R$$ (remember $$R \neq 0$$) to isolate $$h'$$:

$$h' = 4h.$$

This result tells us that the new height has to be four times the original height in order to make the coverage range twice as large. In other words, the height must be multiplied by $$4$$.

Hence, the correct answer is Option B.