In ΔXYZ, z = 92 inches, x = 44 inches and ∠Y=49°. Find the length of y, to the nearest inch.

To solve this problem, we can 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.

Here's how you can apply it:

1. First, find the measure of ∠Z. Since the sum of the angles in a triangle is 180°, you can subtract the given angles from 180° to find ∠Z. So, ∠Z = 180° - 49° - ∠X. But we don't know ∠X yet.

2. To find ∠X, we can use the Law of Sines. According to the Law of Sines, (z/sin∠Z) = (x/sin∠X). We know that z = 92 inches and x = 44 inches. We can plug these values into the equation and solve for sin∠X.

3. sin∠X = (x * sin∠Z) / z = (44 * sin∠Z) / 92.

4. Once we find sin∠X, we can find the measure of ∠X using the inverse sine function.

5. After finding ∠X, we can find ∠Z using the equation from step 1.

6. Finally, we can find y using the Law of Sines again. According to the Law of Sines, (y/sin∠Y) = (z/sin∠Z). We know that sin∠Y = 49° and we have found ∠Z in the previous step. We can plug these values into the equation and solve for y.

7. y = (z * sin∠Y) / sin∠Z = (92 * sin49°) / sin∠Z.

8. Round your answer to the nearest inch.

Please note that this solution assumes that the triangle is not a right triangle. If it is a right triangle, you would use the Pythagorean theorem instead of the Law of Sines.

Step 1: Identify the given values. We have z = 92 inches, x = 44 inches, and ∠Y = 49°.

Step 2: Recognize that this is a case of 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 3: Set up the equation using the Law of Sines. We know two sides and an angle, so we can use the formula to find the other side. The formula is:

a/sinA = b/sinB = c/sinC

In this case, we can set it up as:

x/sinX = z/sinZ

Step 4: We don't know ∠X or ∠Z, but we do know that the sum of the angles in a triangle is 180°. So we can find ∠Z by subtracting ∠Y from 180°:

∠Z = 180° - 49° = 131°

Step 5: Now we can set up our equation:

44/sinX = 92/sin131°

Step 6: We want to find y, which is opposite ∠X. But we don't know ∠X yet. However, we can find it because we know that the angles in a triangle add up to 180°. So:

∠X = 180° - ∠Y - ∠Z = 180° - 49° - 131° = 0°

Step 7: Now we can solve for y:

44/sin0° = 92/sin131°

Step 8: Solving this equation gives us a value of y that is undefined, because sin0° = 0, and division by zero is undefined.

This means that there's an error in the original problem. The given values do not form a valid triangle, because the sum of the two given angles is greater than 180°.