To find the height $h$ of the rectangular prism, we can use the Pythagorean theorem in three dimensions. The diagonal of the rectangular prism (which is given as 18) connects the opposite corners, forming a right triangle with the edges of the prism.

For a rectangular prism with length $a$, width $b$, and height $h$, the length of the space diagonal $d$ can be found using the formula: $d = \sqrt{a^2 + b^2 + h^2}$

Given:

• $a = 2$
• $b = 6$
• $d = 18$

We need to solve for $h$.

Step-by-Step Solution:

1. Substitute the known values into the formula: $18 = \sqrt{2^2 + 6^2 + h^2}$

2. Simplify inside the square root: $18 = \sqrt{4 + 36 + h^2}$ $18 = \sqrt{40 + h^2}$

3. Square both sides to eliminate the square root: $18^2 = 40 + h^2$ $324 = 40 + h^2$

4. Subtract 40 from both sides: $284 = h^2$

5. Take the square root of both sides: $h = \sqrt{284}$

6. Simplify the square root (if possible): $h \approx 16.85$

Thus, the height $h$ of the rectangular prism is approximately 16.85 units.

Would you like further details on this problem, or do you have any questions?

Here are some related questions to explore:

1. How would the formula change if the diagonal of a rectangular prism was unknown?
2. What is the significance of the Pythagorean theorem in three-dimensional shapes?
3. How can we calculate the volume of the rectangular prism with the given dimensions?
4. How do different diagonals within the prism relate to each other?
5. How would we solve this if given only the volume and two dimensions?

Tip: In three-dimensional geometry, space diagonals can often be calculated with the extended Pythagorean theorem for three dimensions.