Final answer:
To determine the number of solutions for the given values, we can use the Law of Sines to find angle B. If the value of angle B is valid, there are two possible solutions for the triangle. If not, there are no solutions.
Step-by-step explanation:
The question asks how many solutions exist for the given values of A, a, and b. To determine the number of solutions, we can use the Law of Sines, which states that for any triangle with sides a, b, and c opposite angles A, B, and C respectively, the following ratio holds: sin A / a = sin B / b = sin C / c. In this case, we have angle A = 29.7°, side a = 27.2 ft, and side b = 41.5 ft. Since angle A is given, we can use the Law of Sines to find angle B.
Using the equation sin A / a = sin B / b, we can substitute the values and solve for sin B: sin 29.7° / 27.2 = sin B / 41.5. Cross-multiplying, we get sin B = (sin 29.7° / 27.2) * 41.5. Taking the arcsin of both sides, we find the value of angle B.
If the value of angle B is a valid angle (between 0° and 180°), then there are two possible solutions for the triangle: one with angle A and angle B, and another with angle A and the supplementary angle of B (180° - angle B). If the value of angle B is not valid, then there are no solutions for the triangle.