The angle of elevation at the top of an unfinished tower at a point distant 78 m from its base is 30°. How much higher does the tower be raised (in m) so that the angle of elevation of the top of the finished tower at the same point will be 60°?

To find how much higher the tower needs to be raised, we can model the situation using right-angled trigonometry.

Let us define the variables based on the problem statement:
Let $B$ be the base of the tower.
Let $A$ be the point of observation, which is at a distance of 78 meters from the base. Therefore, the horizontal distance is:
$AB=78\text{m}$

Let $h_1$ be the height of the unfinished tower, and $h_2$ be the height of the finished tower.

Step 1: Find the height of the unfinished tower ($h_1$)
The angle of elevation of the top of the unfinished tower from point $A$ is $30°$.
Using the tangent trigonometric ratio in the right-angled triangle formed by the unfinished tower:
$\tan (30°)=\frac{\text{Opposite side}}{\text{Adjacent side}}=\frac{h_1}{78}$
Since $\tan (30°)=\frac{1}{\sqrt{3}}$, we can substitute this value:
$\frac{1}{\sqrt{3}}=\frac{h_1}{78}$
Solving for $h_1$:
$h_1=\frac{78}{\sqrt{3}}$
Rationalizing the denominator by multiplying the numerator and denominator by $\sqrt{3}$:
$h_1=\frac{78\sqrt{3}}{3}=26\sqrt{3}\text{m}$

Step 2: Find the height of the finished tower ($h_2$)
The angle of elevation of the top of the finished tower from the same point $A$ is $60°$.
Using the tangent ratio for the finished tower:
$\tan (60°)=\frac{h_2}{78}$
Since $\tan (60°)=\sqrt{3}$, we get:
$\sqrt{3}=\frac{h_2}{78}$
Solving for $h_2$:
$h_2=78\sqrt{3}\text{m}$

Step 3: Calculate how much higher the tower must be raised
The additional height required to finish the tower is the difference between the finished height and the unfinished height:
$\text{Height to be raised}=h_2-h_1$
Substituting the calculated values:
$\text{Height to be raised}=78\sqrt{3}-26\sqrt{3}=52\sqrt{3}\text{m}$

Therefore, the tower must be raised by $52\sqrt{3}$ meters to achieve the desired angle of elevation.