Answer:
The crate weighs approximately 1082.21 N.
Step-by-step explanation:
To determine the weight of the crate, we need to calculate the vertical components of the tension forces in both ropes and sum them up. This is because the vertical components of the tension forces will balance the weight of the crate when it's in equilibrium.
For the first rope with a tension of 738 N at an angle of 31° with the horizontal, the vertical component \( T_{1y} \) is calculated as:
\[ T_{1y} = 738 \times \sin(31°) \]
For the second rope with a tension of 945 N at an angle of 48° with the horizontal, the vertical component \( T_{2y} \) is calculated as:
\[ T_{2y} = 945 \times \sin(48°) \]
The total weight \( W \) of the crate is the sum of these two vertical components:
\[ W = T_{1y} + T_{2y} \]
Now, let's calculate the values:
\[ T_{1y} = 738 \times \sin(31°) \approx 738 \times 0.515 \approx 380.07 \, N \]
\[ T_{2y} = 945 \times \sin(48°) \approx 945 \times 0.743 \approx 702.14 \, N \]
\[ W = 380.07 \, N + 702.14 \, N \approx 1082.21 \, N \]
Therefore, the crate weighs approximately **1082.21 N**.
