1. Use the formula for work done

For a force $F$ acting at an angle $\theta$ to the displacement $d$,

$W = Fd\cos\theta$

2. Write work done by A and B

Since both do the same work over the same displacement $d$:

• For person A, angle $=45^\circ$ $W_A = F_A d \cos 45^\circ$

• For person B, angle $=60^\circ$ $W_B = F_B d \cos 60^\circ$

Given $W_A = W_B$,

$F_A d \cos 45^\circ = F_B d \cos 60^\circ$

Cancel $d$:

$F_A \cos 45^\circ = F_B \cos 60^\circ$

3. Substitute trigonometric values

$\cos 45^\circ = \frac{1}{\sqrt{2}}, \qquad \cos 60^\circ = \frac{1}{2}$

So,

$F_A \cdot \frac{1}{\sqrt{2}} = F_B \cdot \frac{1}{2}$

Hence,

$\frac{F_A}{F_B} = \frac{\frac{1}{2}}{\frac{1}{\sqrt{2}}} = \frac{\sqrt{2}}{2} = \frac{1}{\sqrt{2}}$

4. Compare with given ratio

Given,

$\frac{F_A}{F_B} = \frac{1}{\sqrt{x}}$

Thus,

$\frac{1}{\sqrt{x}} = \frac{1}{\sqrt{2}}$

Therefore,

$x=2$

5. Comparison with stored answer

Stored correct answer is $2$, which matches the derived answer.