Interpret the geometry

The force $\vec P$ of magnitude $100\,\text{N}$ acts vertically downward at point $A$.

It is to be resolved along the two non-perpendicular arms $AB$ and $AC$.

From the given trigonometric values, the included geometry is consistent with:

• arm $AB$ making $35^\circ$ with the vertical,
• arm $AC$ making $35^\circ$ with the horizontal on the other side, so the angle between the two arms is $\angle BAC = 110^\circ.$

Hence, when resolving a force along two non-orthogonal directions, we use the triangle/law of sines method.

Let the resolved components be

• along $AB$: $F_{AB}$
• along $AC$: $F_{AC}=x$

Their vector sum equals the applied force $P=100\,\text{N}$.

Use the law of sines for resolution along two inclined directions

If a force $P$ is resolved along two directions with included angle $110^\circ$, and the angle between $P$ and one arm is $35^\circ$, then the angle between $P$ and the other arm is also $180^\circ - 110^\circ - 35^\circ = 35^\circ.$

For oblique resolution: $\frac{F_{AC}}{\sin 35^\circ} = \frac{P}{\sin 110^\circ}$

Therefore, $F_{AC} = P\,\frac{\sin 35^\circ}{\sin 110^\circ}.$

Substitute the given values

$F_{AC} = 100\times \frac{0.573}{0.939}$

$F_{AC} \approx 100\times 0.6102 = 61.02\,\text{N}$

So, $x \approx 61\,\text{N}.$

Nearest integer

$\boxed{x=61}$

Compare with stored answer

Stored correct answer = $82$.

My derived answer is $61$, which does not match $82$.

The likely reason is that the actual figure may have a different placement of the $35^\circ$ angle than inferred from the text alone. If instead the component along $AC$ is opposite the $35^\circ$ angle in the force triangle, one would get $x = 100\frac{\sin 110^\circ}{\sin(\text{other angle})},$ which can produce a value near $82$ depending on the exact figure. However, from the standard oblique resolution with the most natural interpretation of the supplied angles, the result is $61\,\text{N}$.

Therefore I disagree with the stored answer based on the information visible in the prompt alone.