Find all solutions to the following triangle. (Round your answers to the nearest whole number. If either triangle is not possible, enter NONE in each corresponding answer blank.)A = 32°, a = 44 ft, b = 52 ftFirst triangle (assume B ≤ 90°):B  =  °C  =

To solve this problem, we will use the Law of Sines, which states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all three sides of the triangle.

Step 1: Find angle B We can start by finding angle B using the Law of Sines. We know that a/sin(A) = b/sin(B), so we can plug in the given values and solve for B:

44/sin(32) = 52/sin(B)

Solving for sin(B) gives us sin(B) = 52*sin(32)/44 = 0.745

Taking the inverse sine of both sides gives us B = sin^-1(0.745) = 48 degrees (rounded to the nearest whole number).

Step 2: Find angle C We know that the sum of the angles in a triangle is 180 degrees, so we can find angle C by subtracting angles A and B from 180:

C = 180 - 32 - 48 = 100 degrees

So, the solutions for the first triangle are B = 48 degrees and C = 100 degrees.

However, since the sum of angles A and B is less than 90 degrees, there could be another possible triangle.

Step 3: Find the second possible angle B Since sin(B) = sin(180 - B), the second possible angle B is B = 180 - 48 = 132 degrees.

Step 4: Find the second possible angle C Similarly, we can find the second possible angle C by subtracting angles A and B from 180:

C = 180 - 32 - 132 = 16 degrees

So, the solutions for the second triangle are B = 132 degrees and C = 16 degrees.

However, since angle B is greater than 90 degrees, this triangle is not possible according to the problem's instructions. Therefore, the solutions for the second triangle are NONE.