Determine the remaining sides and angles of the triangle ABC. \[ a=100 m, A=39^{\circ} 54^{\prime}, C=27^{\circ} 26^{\prime} \] What is the measure of angle B? \[ B=.112^{\circ} 40^{\prime} \] What is the length of side $b$ ? \[ \mathrm{b}=\square \mathrm{m} \] (Do not round until the final answer. Then round to the nearest meter as needed.)

Step 1 ：We are given that the lengths of side a is 100 m, and the measures of angles A and C are \(39^{\circ} 54^{\prime}\) and \(27^{\circ} 26^{\prime}\) respectively.

Step 2 ：We know that the sum of the angles in a triangle is \(180^{\circ}\). So, we can find the measure of angle B by subtracting the measures of angles A and C from \(180^{\circ}\).

Step 3 ：Converting the given angles to decimal form, we get A = 39.9 and C = 27.433333333333334.

Step 4 ：Subtracting these from 180, we get B = \(180 - 39.9 - 27.433333333333334 = 112.66666666666666^{\circ}\).

Step 5 ：Next, we can use the law of sines to find the length of side b. The law of sines 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 6 ：First, we convert the angles from degrees to radians. We get A_rad = 0.6963863715457375, B_rad = 1.9664042905802779, and C_rad = 0.4788019914637777.

Step 7 ：Then, we use the law of sines to find b: \(b = a \cdot \frac{\sin(B)}{\sin(A)}\). Substituting the given values, we get \(b = 100 \cdot \frac{\sin(1.9664042905802779)}{\sin(0.6963863715457375)} = 144\).

Step 8 ：So, the length of side b is \(\boxed{144}\) meters.