Question

Two forces with magnitudes of 25 and 30 pounds act on an object at angles of 10° and 100° respectively. Find the direction and magnitude of the resultant force. Round to two decimal places in all intermediate steps and in your final answer.

Help me please

Answer 1

Answer on Question #42430 – Math - Analytic Geometry

Two forces with magnitudes of 25 and 30 pounds act on an object at angles of $10{}^{\circ}$ and $100{}^{\circ}$ respectively. Find the direction and magnitude of the resultant force. Round to two decimal places in all intermediate steps and in your final answer.

Help me please

Solution.

$$OA = 25, \quad OB = 30, \angle DOA = 10{}^{\circ}, \quad \angle DOB = 100{}^{\circ},$$ $$\angle BOA = \angle BCA = 100{}^{\circ} - 10{}^{\circ} = 90{}^{\circ}, \text{so } \angle OAC = \angle OBC = 90{}^{\circ}.$$

Therefore $OC = \sqrt{OA^2 + AC^2} = \sqrt{OA^2 + OB^2} = \sqrt{25^2 + 30^2} = \sqrt{1525} \cong 39.05$

$$\tan(\angle COA) = \frac{AC}{OA} = \frac{30}{25} = 1.2 \rightarrow \angle COA = \arctan(1.2) = 50.19{}^{\circ}.$$ $$\angle COD = \angle COA + \angle AOD = 50.19{}^{\circ} + 10{}^{\circ} = 60.19{}^{\circ}.$$

Answer: magnitude of the resulting force equals 39.05 pounds, direction of the resulting force equals $60.19{}^{\circ}$.

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