1. The problem states: Two forces of magnitudes $3$ and $5$ newton act at a point with an included angle of $60^\circ$. Find the magnitude of their resultant force. 2. The formula for the magnitude of the resultant $R$ of two forces $F_1$ and $F_2$ acting at an angle $\theta$ is: $$ R = \sqrt{F_1^2 + F_2^2 + 2 F_1 F_2 \cos \theta} $$ 3. Substitute the given values: $F_1 = 3$, $F_2 = 5$, and $\theta = 60^\circ$: $$ R = \sqrt{3^2 + 5^2 + 2 \times 3 \times 5 \times \cos 60^\circ} $$ 4. Calculate each term: $$ 3^2 = 9, \quad 5^2 = 25, \quad \cos 60^\circ = \frac{1}{2} $$ 5. Substitute these values: $$ R = \sqrt{9 + 25 + 2 \times 3 \times 5 \times \frac{1}{2}} $$ 6. Simplify the multiplication inside the square root: $$ 2 \times 3 \times 5 \times \frac{1}{2} = \cancel{2} \times 3 \times 5 \times \frac{1}{\cancel{2}} = 3 \times 5 = 15 $$ 7. So, $$ R = \sqrt{9 + 25 + 15} = \sqrt{49} $$ 8. Finally, $$ R = 7 $$ Therefore, the magnitude of the resultant force is $7$ newton. Answer: (c) $7$