The problem asks to find the height $h$ of the rectangular prism. The diagram shows the dimensions of the base (6 and 3) and the diagonal of the prism (9).

To solve for $h$, we use the Pythagorean theorem in three dimensions. The diagonal $d$ of a rectangular prism is given by:

$d = \sqrt{l^2 + w^2 + h^2}$

Where:

• $l = 6$ (length),
• $w = 3$ (width),
• $d = 9$ (diagonal).

Substituting the known values:

$9 = \sqrt{6^2 + 3^2 + h^2}$

Simplify:

$9 = \sqrt{36 + 9 + h^2}$

$9 = \sqrt{45 + h^2}$

Square both sides:

$81 = 45 + h^2$

$h^2 = 81 - 45$

$h^2 = 36$

Take the square root of both sides:

$h = \sqrt{36}$

$h = 6$

Thus, the height $h$ of the rectangular prism is $\mathbf{6}$.

Would you like a detailed explanation of any part of this solution?

Here are five related questions to explore further:

1. What is the formula for the diagonal of a rectangular prism and how can it be derived?
2. How would the solution change if the diagonal was a different length?
3. What is the volume of this rectangular prism given these dimensions?
4. How is the Pythagorean theorem extended into three dimensions?
5. Could you verify the solution by recalculating the diagonal using $h = 6$?

Tip: Always ensure you check your work by substituting the solution back into the original formula.