Step 1: Understand the Geometry

Let the height of the tower be $AB = 10\text{ m}$. Let the two roads be at points $C$ and $D$ on opposite sides of the tower on the ground. The angles of depression from the top of the tower $A$ to roads $C$ and $D$ are $30°$ and $45°$ respectively, which means the angles of elevation from the roads to the top of the tower are also $30°$ and $45°$.

$h = 10\text{ m}, \theta_1 = 30°, \theta_2 = 45°$

Step 2: Calculate Distance to the First Road

In right-angled triangle $ABC$, using the tangent trigonometric ratio: \tan 30°= \frac{AB}{BC} = \frac{10}{BC}BC = \frac{10}{\tan 30° = 10\sqrt{3} \approx 10 \times 1.732 = 17.32\text{ m}

$BC = 17.32\text{ m}$

Step 3: Calculate Distance to the Second Road

In right-angled triangle $ABD$, using the tangent trigonometric ratio: \tan 45°= \frac{AB}{BD} = \frac{10}{BD}BD = \frac{10}{\tan 45° = \frac{10}{1} = 10.00\text{ m}

$BD = 10.00\text{ m}$

Step 4: Find Total Distance Between Roads

Since the roads are on opposite sides of the tower, the total distance between the roads is the sum of the two distances $BC$ and $D$: $\text{Total Distance} = BC + BD = 17.32 + 10 = 27.32\text{ m}$

$\text{Total Distance} = 27.32\text{ m}$