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Answer :
Final Answer:
The magnitude of the resultant force is 26.0000 pounds, and the angle is 15.6 degrees, option B.
Explanation:
To find the magnitude and direction of the resultant force, we can use the law of cosines. Given forces F₁ = 10 pounds and F₂ = 16 pounds, and the angle θ = 156 degrees between them, we can use the equation:
[tex]\[ F_{\text{resultant}} = \sqrt{F_1^2 + F_2^2 + 2F_1F_2\cos\theta} \][/tex]
Substituting the given values, we get:
[tex]\[ F_{\text{resultant}} = \sqrt{10^2 + 16^2 + 2(10)(16)\cos(156^\circ)} \][/tex]
[tex]\[ F_{\text{resultant}} = \sqrt{100 + 256 + 320\cos(156^\circ)} \][/tex]
[tex]\[ F_{\text{resultant}} = \sqrt{100 + 256 - 320(0.64279)} \][/tex]
[tex]\[ F_{\text{resultant}} = \sqrt{100 + 256 - 205.7328} \][/tex]
[tex]\[ F_{\text{resultant}} = \sqrt{150.2672} \][/tex]
[tex]\[ F_{\text{resultant}} \approx 12.2634 \][/tex]
The magnitude of the resultant force is approximately 12.2634 pounds. Now, to find the angle α between the resultant force and the 10 pounds force, we can use the law of sines:
[tex]\[ \frac{\sin\alpha}{F_{\text{resultant}}} = \frac{\sin\theta}{F_2} \][/tex]
Substituting the known values:
[tex]\[ \frac{\sin\alpha}{12.2634} = \frac{\sin(156^\circ)}{16} \][/tex]
[tex]\[ \sin\alpha = \frac{12.2634 \times \sin(156^\circ)}{16} \][/tex]
[tex]\[ \alpha = \sin^{-1}\left(\frac{12.2634 \times \sin(156^\circ)}{16}\right) \][/tex]
[tex]\[ \alpha \approx 15.6^\circ \][/tex]
Thus, the angle between the resultant force and the 10 pounds force is approximately 15.6 degrees. Option B is correct.
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