Correct Answer: B ($4(3 + \sqrt{3})m, 4(3 + \sqrt{3})m$)

This problem can be solved by setting up a system of two equations using the trigonometric ratios from the two right-angled triangles formed by the buildings and the lines of sight.

Step 1: Set up the Geometry and Variables

Let's define the components of the problem:

• Let H be the height of the multi-storied building.
• Let x be the horizontal distance between the two buildings.
• The height of the shorter building is 8m.

We can form two right-angled triangles based on the given angles of depression:

1. A triangle involving the bottom of the 8m building.
2. A triangle involving the top of the 8m building.

By the property of alternate interior angles, the angle of depression from the top of the tall building to a point is equal to the angle of elevation from that point to the top of the tall building.

Step 2: Formulate Equations using Trigonometry

Triangle 1: Using the angle of depression to the bottom ($45^\circ$)

This triangle involves the full height of the multi-storied building (H) and the horizontal distance (x).

$$ \tan(45^\circ) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{H}{x} $$

Since $\tan(45^\circ) = 1$, we get:

$$ 1 = \frac{H}{x} \implies H = x $$

This tells us that the height of the multi-storied building is equal to the distance between the buildings.

Triangle 2: Using the angle of depression to the top ($30^\circ$)

This triangle involves the horizontal distance (x) and the portion of the tall building's height that is above the shorter building. This height difference is $H - 8$.

$$ \tan(30^\circ) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{H - 8}{x} $$

Step 3: Solve the System of Equations

We have two equations:

1. $H = x$
2. $\tan(30^\circ) = \frac{H-8}{x}$

Substitute the first equation into the second one, and use the value $\tan(30^\circ) = \frac{1}{\sqrt{3}}$:

$$ \frac{1}{\sqrt{3}} = \frac{H - 8}{H} $$

Now, we solve for H:

$H = \sqrt{3}(H - 8)$
$H = H\sqrt{3} - 8\sqrt{3}$
$8\sqrt{3} = H\sqrt{3} - H$
$8\sqrt{3} = H(\sqrt{3} - 1)$
$H = \frac{8\sqrt{3}}{\sqrt{3} - 1}$

To rationalize the denominator, multiply the numerator and denominator by the conjugate, $(\sqrt{3} + 1)$:

$H = \frac{8\sqrt{3}(\sqrt{3} + 1)}{(\sqrt{3} - 1)(\sqrt{3} + 1)} = \frac{8(3 + \sqrt{3})}{3 - 1} = \frac{8(3 + \sqrt{3})}{2}$

$H = 4(3 + \sqrt{3})$

Step 4: State the Final Answer

We found the height of the multi-storied building to be $H = 4(3 + \sqrt{3})$ m.

Since we established in Step 2 that $H = x$, the distance between the two buildings is also $x = 4(3 + \sqrt{3})$ m.