Me i will mark ! 1. - two forces with magnitudes of 90 and 50 pounds act on an object at angles of 30° and 160°, respectively. find the direction and magnitude of the resultant force. round to two decimal places in all intermediate steps and in your final answer. 2. - find the fourth roots of 256(cos 240° + i sin 240°).

1. One way to approach this is to convert the magnitude and direction of each force to rectangular coordinates.

... first force = 90(cos(30°), sin(30°)) = (77.94, 45.00)

... second force = 50(cos(160°), sin(160°)) = (-46.98, 17.10)

Then their sum is

... resultant force = first force + second force = (30.96, 62.10)

The magnitude is found using the Pythagorean theorem:

... resultant magnitude = √(30.96^2 + 62.10^2)

... ... = √(958.52+3856.41) = √4814.93

... ... = 69.39

The resultant direction can be found using the definition of the tangent function.

... tan(α) = 62.10/30.96 = 2.01

... α = arctan(2.01) = 63.55°

The (magnitude, direction) of the resultant force are (69.39, 63.55°).

(Note that rounding introduced an error of 0.05° in the direction.)

2. Using DeMoivre's theorem, the magnitude of the 4th root is the 4th root of the magnitude, and the angle of the 4th root is 1/4 of the angle (and its aliases).

... fourth root = 256^(1/4)∠((240°+k*360°)/4) = 4∠(60°+k*90°) . . . . k=0, 1, 2, 3

(a) the volume of the cone is 20/3 in3

hence, option a is correct.