Let’s analyze the problem and solve it.

Step 1: Understanding the diagram

The figure shows a cuboid with the following knowns:

• $AD = 63 \, \text{mm}$ (along the base).
• We are asked to find the length of the space diagonal $AG$, which stretches from corner $A$ to the opposite corner $G$.

There are two given angles:

• $\angle DAB = 42^\circ$ (blue angle).
• $\angle BAE = 27^\circ$ (orange angle).

Our goal is to use these angles and the known side $AD$ to calculate the length of $AG$ to 2 decimal places.

Step 2: Breaking down the triangles

Triangle $ABD$:

We first need to determine the length of the diagonal $AB$, which lies on the base of the cuboid. In right triangle $ABD$, the angle at $A$ is $42^\circ$, and the side adjacent to it is $AD = 63 \, \text{mm}$.

Using the trigonometric relation for cosine: $\cos(42^\circ) = \frac{AD}{AB}$ $AB = \frac{AD}{\cos(42^\circ)} = \frac{63}{\cos(42^\circ)}$

Now, calculate: $\cos(42^\circ) \approx 0.7431$ $AB = \frac{63}{0.7431} \approx 84.78 \, \text{mm}$

Triangle $ABE$:

Next, we use right triangle $ABE$ to find the space diagonal $AG$. The angle $\angle BAE = 27^\circ$, and the side adjacent to it is $AB = 84.78 \, \text{mm}$.

Using the relation for cosine again: $\cos(27^\circ) = \frac{AB}{AG}$ $AG = \frac{AB}{\cos(27^\circ)}$

Now, calculate: $\cos(27^\circ) \approx 0.8910$ $AG = \frac{84.78}{0.8910} \approx 95.17 \, \text{mm}$

Step 3: Final Answer

The length of $AG$ is: $\boxed{95.17 \, \text{mm}}$

Would you like further clarifications or more details on any step?

Here are five follow-up questions:

1. How would the calculation change if the base side $AD$ were different?
2. What is the formula for a space diagonal in terms of the cuboid's dimensions?
3. Can we solve for $AG$ without using trigonometry?
4. How can we use vectors to find the length of $AG$?
5. What if $\angle DAB$ were larger—how would that affect the result?

Tip: When working with multiple right triangles, identify the intermediate values (like diagonal lengths) step-by-step to avoid confusion.