The shadow of a tower standing on a level ground is found to be 40 m longer when the Sun's altitude is $30^\circ$ than when it is $60^\circ$. Find the height of the tower.

Step 1: Define variables and set up the problem

We first define the variables for the height of the tower and the lengths of the shadows. The problem states that the shadow is 40 m longer when the altitude is $30^\circ$ compared to when it is $60^\circ$. This gives us two right-angled triangles, $\triangle ABC$ and $\triangle ABD$.

Step 2: Apply trigonometry in $\triangle$ ABC

In the right-angled triangle $\triangle ABC$, the angle of elevation is $60^\circ$. We use the tangent function, which relates the opposite side (height of the tower $h$) to the adjacent side (length of the shadow $x$). We know that $\tan 60^\circ = \sqrt{3}$.

Step 3: Express 'h' in terms of 'x'

From the previous step's equation, we can express the height of the tower $h$ in terms of $x$. This relationship will be used to solve for $x$ in the subsequent steps.

Step 4: Apply trigonometry in $\triangle$ ABD

In the right-angled triangle $\triangle ABD$, the angle of elevation is $30^\circ$. The opposite side is $h$ and the adjacent side is $x + 40$. We know that $\tan 30^\circ = \frac{1}{\sqrt{3}}$.

Step 5: Substitute 'h' and solve for 'x'

Now we substitute the expression for $h$ from Step 3 into the equation from Step 4. This allows us to form an equation solely in terms of $x$. We then solve this linear equation to find the value of $x$, which represents the shorter shadow length.

Step 6: Calculate the height of the tower 'h'

With the value of $x$ found, we can now substitute it back into the equation for $h$ derived in Step 3. This gives us the height of the tower in terms of $\sqrt{3}$.

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