Final answer:
To find the magnitude of the smaller force, you can use the law of cosines. Rearrange the equation to solve for F2 using the quadratic formula. Plug in the value of F1 given in the problem to calculate F2 to the nearest 10th of a pound.
Step-by-step explanation:
To find the magnitude of the smaller force, we can use the law of cosines. Let's denote the magnitude of the larger force as F1 and the magnitude of the smaller force as F2. The angle between the resultant and F1 is 56 degrees, and the angle between the resultant and F2 is 8 degrees.
The law of cosines states that the square of the magnitude of the resultant is equal to the sum of the squares of the magnitudes of the individual forces minus twice the product of the magnitudes of the forces and the cosine of the angle between them.
Using the given values, we have:
R^2 = F1^2 + F2^2 - 2 * F1 * F2 * cos(56 - 8)
Substituting the known values, we get:
63^2 = F1^2 + F2^2 - 2 * F1 * F2 * cos(48)
To solve for F2, rearrange the equation:
F2^2 - 2 * F1 * F2 * cos(48) + F1^2 - 63^2 = 0
This is a quadratic equation in terms of F2. Solve for F2 using the quadratic formula:
F2 = (-(-2 * F1 * cos(48)) ± sqrt((-2 * F1 * cos(48))^2 - 4 * 1 * (F1^2 - 63^2))) / 2
Choose the positive value for F2 since force values cannot be negative. Plug in the value of F1 given in the problem, and calculate F2 to the nearest 10th of a pound.