This is a right triangle problem involving trigonometry. To solve for $x$, we will first need to calculate the length of the "middle line," which is the side opposite the 55° angle and adjacent to the 24.1' hypotenuse.

Let's break it down step by step:

1. Step 1: Calculate the middle line using trigonometry.

   • Use the sine function, since the 24.1' is the hypotenuse and the "middle line" is opposite the 55° angle.

   $\sin(55^\circ) = \frac{\text{opposite (middle line)}}{\text{hypotenuse}}$ Therefore, $\text{middle line} = 24.1' \times \sin(55^\circ)$

2. Step 2: Use the cosine rule to find $x$.

   • Now that we have the length of the middle line, we can use the cosine law for the right triangle formed by the angles 47° and 55° to find $x$.

Let me calculate these values for you.The length of the middle line (opposite the 55° angle) is approximately 19.74 units. Using this, the value of $x$ is approximately 28.95 when rounded to two decimal places.

Would you like further details or explanations on any step?

Here are 5 related questions to deepen your understanding:

1. How does the sine function help in finding the length of a side in a right triangle?
2. What other trigonometric identities could be used to solve similar triangle problems?
3. How does the cosine function relate to finding side lengths in right triangles?
4. What is the significance of the right angle in this triangle for solving with trigonometry?
5. How would the problem change if a different angle were given?

Tip: In trigonometry, always double-check that your calculator is in the correct mode (degrees vs radians) based on the angle measurements you are using!