To calculate the length of $AG$ in the cuboid, we can use trigonometric relationships and the Pythagorean theorem. Here’s a step-by-step approach:

1. Calculate the length of $AC$:

$AC$ is the diagonal of the base rectangle $ABCD$, which has sides $AB = 33 \, \text{cm}$ and $AD = 21 \, \text{cm}$. Using the Pythagorean theorem: $AC = \sqrt{AB^2 + AD^2}$ Substitute the values: $AC = \sqrt{33^2 + 21^2} = \sqrt{1089 + 441} = \sqrt{1530}$ $AC \approx 39.11 \, \text{cm}.$

2. Calculate the length of $AG$:

$AG$ is the diagonal of the cuboid from $A$ to $G$, and spans the length $AC$ and height $AH = 72 \, \text{cm}$. Again, using the Pythagorean theorem: $AG = \sqrt{AC^2 + AH^2}$ Substitute the values: $AG = \sqrt{39.11^2 + 72^2} = \sqrt{1529.93 + 5184} = \sqrt{6713.93}$ $AG \approx 81.94 \, \text{cm}.$

Final Answer:

$AG \approx 81.94 \, \text{cm} \, \text{(to 2 decimal places)}.$

Would you like me to clarify any step further or calculate other related dimensions?

Questions to Explore:

1. How do you calculate the diagonal of a rectangular prism in general?
2. What are other scenarios where the Pythagorean theorem is useful in 3D geometry?
3. Could trigonometric functions have simplified this calculation further?
4. How can you confirm your answers with a diagram?
5. What happens if you change one dimension of the cuboid?

Tip:

Always ensure that the dimensions (e.g., lengths and angles) are correctly identified and assigned to their respective sides before applying any formulas. This avoids mix-ups and incorrect solutions!