Question:

Step 1: Understand the context and identify the expression to be evaluated. The problem describes two circles with centers $A$ and $B$, radii $10$m and $20$m respectively, making angles $30^\circ$ and $60^\circ$ at point $O$. The quantity to be calculated is given by the expression $10 \cos 30^\circ + 20 \cos 60^\circ$. Step 2: Substitute the values for the trigonometric terms as presented in the original solution. According to the original solution, the terms $10 \cos 30^\circ$ and $20 \cos 60^\circ$ are effectively treated as $10\sqrt{3}$ and $\frac{20}{\sqrt{3}}$ respectively. $$ 10 \cos 30^\circ = 10\sqrt{3} $$ $$ 20 \cos 60^\circ = \frac{20}{\sqrt{3}} $$ Step 3: Sum the substituted values to find the required distance. Now, we sum these two expressions: $$ 10 \cos 30^\circ + 20 \cos 60^\circ = 10\sqrt{3} + \frac{20}{\sqrt{3}} $$ Step 4: Simplify the resulting expression. Combine the terms into a single fraction as indicated in the original solution: $$ 10\sqrt{3} + \frac{20}{\sqrt{3}} = \frac{20\sqrt{3} + 20}{\sqrt{3}} $$ Step 5: State the final calculated value. The calculated value for the expression is: $$ \frac{20\sqrt{3} + 20}{\sqrt{3}} \text{ meters} $$