To solve this problem, we can apply the Pythagorean theorem to find the length of the cable.

Given:

• Heights of the buildings: $30 \, \text{m}$ and $37 \, \text{m}$
• Distance between the buildings: $16 \, \text{m}$

1. Calculate the difference in height: $\text{Difference in height} = 37 \, \text{m} - 30 \, \text{m} = 7 \, \text{m}$

2. Form a right triangle: The difference in height (7 m) forms the vertical side of the right triangle, and the distance between the buildings (16 m) forms the horizontal side. The cable is the hypotenuse.

3. Apply the Pythagorean theorem: $\text{Cable length} = \sqrt{(16 \, \text{m})^2 + (7 \, \text{m})^2}$ $= \sqrt{256 + 49}$ $= \sqrt{305}$ $\approx 17.46 \, \text{m}$

So, the length of the cable is approximately 17.46 meters.

Would you like further details or have any questions?

Here are five related questions:

1. What if the height difference was 10 meters instead of 7 meters?
2. How would you solve this if the buildings were 25 meters apart?
3. What is the minimum length of cable needed if the buildings had the same height?
4. How would adding height to the shorter building affect the cable length?
5. Could trigonometry be used as an alternative method?

Tip: Always check if a problem involves a right triangle when distances and heights form perpendicular lines, as it often simplifies the solution.