We multiply the area by 2 to solve for $r$ because of the formula used to calculate the area of the triangle. The formula for the area of a triangle using two sides and the included angle is:

$\text{Area} = \frac{1}{2}ab \sin C$

In our problem:

• $a$ and $b$ are the distances $OB$ and $OC$, which are both equal to $r$.
• $C$ is the angle between $OB$ and $OC$, which is $62^\circ$.

Thus, the formula becomes:

$\text{Area} = \frac{1}{2} r^2 \sin 62^\circ$

Given that the area is 148 km², we set up the equation:

$148 = \frac{1}{2} r^2 \sin 62^\circ$

To isolate $r^2$, we multiply both sides of the equation by 2:

$2 \times 148 = r^2 \sin 62^\circ$

$296 = r^2 \sin 62^\circ$

Therefore, the multiplication by 2 is necessary to eliminate the fraction in the equation, making it possible to solve for $r$.

Would you like more clarification or have any further questions?

Additional Questions:

1. How is the formula for the area of a triangle derived?
2. Can this method be applied to non-right triangles? Why or why not?
3. What is the relationship between the sine function and the area of a triangle?
4. What other trigonometric functions can be used to find the area of a triangle?
5. How does the value of the sine function change with different angles?
6. What would the effect be if the angle $C$ were obtuse?
7. Why do we use radians instead of degrees in some calculations?
8. How would the calculation change if the area were given in different units?

Tip:

Always remember to double-check your work when converting between different trigonometric function representations, such as degrees and radians, as mistakes can easily lead to incorrect calculations.