To find the required length of the hose, we need to determine the distance between opposite corners of the rectangular garden bed, which measures 12 feet by 9 feet.

This is essentially the length of the diagonal of a rectangle, which can be found using the Pythagorean theorem. The diagonal $d$ of a rectangle with side lengths $a$ and $b$ is given by:

$d = \sqrt{a^2 + b^2}$

Calculation:

1. Here, $a = 12$ feet and $b = 9$ feet.

2. Substitute these values into the formula:

   $d = \sqrt{12^2 + 9^2} = \sqrt{144 + 81} = \sqrt{225}$

3. Simplify the square root:

   $d = 15 \text{ feet}$

Answer:

The required length of the hose is 15 feet.

Would you like more details on the steps involved or have any further questions?

Here are some related questions for further practice:

1. How would the length of the hose change if the garden bed was 15 feet by 20 feet?
2. What would be the diagonal length if the garden was square with each side measuring 10 feet?
3. If the garden bed was a right triangle instead of a rectangle, how would you find the hose length to reach the opposite corner?
4. How does the Pythagorean theorem apply to finding distances in a 3D space?
5. What would the answer be if the hose needed to be measured in inches instead of feet?

Tip: Remember, the Pythagorean theorem is a helpful tool whenever you need to find the diagonal of a rectangular shape!