One boat pulls a barge with a force of 100 newtons. Another boat pulls the barge at an angle of 45° to the first force, with a force of 200 newtons. Find the resultant force acting on the barge, to the nearest unit, and the angle between the resultant and the first boat, to the nearest tenth.

Verified step by step guidance

1. Identify the two forces acting on the barge: \( F_1 = 100 \) newtons and \( F_2 = 200 \) newtons, with an angle of \( 45^\circ \) between them.
2. Use the Law of Cosines to find the magnitude of the resultant force \( R \). The formula is: \[ R = \sqrt{F_1^2 + F_2^2 + 2 F_1 F_2 \cos(\theta)} \] where \( \theta = 45^\circ \).
3. Substitute the known values into the formula: \[ R = \sqrt{100^2 + 200^2 + 2 \times 100 \times 200 \times \cos(45^\circ)} \].
4. To find the angle \( \alpha \) between the resultant force and the first force \( F_1 \), use the Law of Sines or the formula derived from the Law of Cosines: \[ \alpha = \arccos \left( \frac{F_1^2 + R^2 - F_2^2}{2 F_1 R} \right) \].
5. Substitute the values of \( F_1 \), \( F_2 \), and the resultant \( R \) into the angle formula to calculate \( \alpha \), which gives the angle between the resultant force and the first boat's force.

Key Concepts

Vector Addition

Vector addition involves combining two or more vectors to find a resultant vector. Since forces have both magnitude and direction, adding them requires considering their directions, often using components or the parallelogram law to find the total force acting on an object.

Law of Cosines

The Law of Cosines relates the lengths of sides of a triangle to the cosine of one of its angles. It is useful for finding the magnitude of the resultant vector when two vectors form an angle, by treating the vectors as sides of a triangle and calculating the third side.

Angle Between Vectors

Determining the angle between the resultant vector and one of the original vectors involves using trigonometric ratios or the Law of Sines. This helps in understanding the direction of the resultant force relative to a reference vector.