Solve the triangle. (Round your answer for side b to the nearest whole number. Round your answers for angles A and C to one decimal place.)a = 409 m, c = 339 m, B = 153.5°b  =  mA  =  °C  =  °

To solve the triangle, we can use the Law of Sines and the Law of Cosines.

Step 1: Use the Law of Cosines to find side b. The Law of Cosines is a^2 = b^2 + c^2 - 2bc cos(A). But we don't know angle A, so we rearrange the formula to solve for b: b = sqrt(a^2 + c^2 - 2ac cos(B)).

Substitute the given values into the formula: b = sqrt((409)^2 + (339)^2 - 2409339 cos(153.5°)) b = sqrt(167281 + 114921 - 277986 cos(153.5°)) b = sqrt(282202 - 277986 cos(153.5°)) b = sqrt(282202 - (-267,000)) b = sqrt(549202) b = 741 m (rounded to the nearest whole number)

Step 2: Use the Law of Sines to find angle A. The Law of Sines is a/sin(A) = b/sin(B). Rearrange the formula to solve for A: A = arcsin(a sin(B) / b).

Substitute the given values into the formula: A = arcsin(409 sin(153.5°) / 741) A = arcsin(409 * -0.4226 / 741) A = arcsin(-173.1 / 741) A = arcsin(-0.2336) A = -13.6°

Since the angle cannot be negative, we add 180° to get the correct angle in the second quadrant: A = -13.6° + 180° A = 166.4° (rounded to one decimal place)

Step 3: Use the fact that the sum of the angles in a triangle is 180° to find angle C. C = 180° - A - B C = 180° - 166.4° - 153.5° C = -140°

Since the angle cannot be negative, we add 180° to get the correct angle in the second quadrant: C = -140° + 180° C = 40° (rounded to one decimal place)

So, the solutions are b = 741 m, A = 166.4°, and C = 40°.