To find the length of $AG$ in the cuboid, let's break this down step-by-step:

Step 1: Understand the geometry of the cuboid

• Given dimensions:

  • Length of $AD = 23 \, \text{cm}$ (base along the bottom left edge).
  • Width of $DC = 18 \, \text{cm}$ (base along the bottom right edge).
  • $\angle DAB = 75^\circ$ (angle between $AD$ and $AB$).

  The diagonal $AG$ runs from $A$ at the bottom front corner to $G$, the farthest opposite top corner.

Step 2: Find $AB$ using trigonometry

From $\angle DAB = 75^\circ$ and the length of $AD = 23 \, \text{cm}$, we can calculate $AB$ using: $AB = AD \cdot \tan(75^\circ)$ Substitute: $AB = 23 \cdot \tan(75^\circ)$ Using $\tan(75^\circ) \approx 3.732$: $AB \approx 23 \cdot 3.732 = 85.84 \, \text{cm}$

Step 3: Find $AC$, the diagonal of the base

The diagonal $AC$ can be found using the Pythagorean theorem, as $AC$ is the hypotenuse of the triangle $\triangle ADC$: $AC = \sqrt{AD^2 + DC^2}$ Substitute: $AC = \sqrt{23^2 + 18^2} = \sqrt{529 + 324} = \sqrt{853}$ $AC \approx 29.23 \, \text{cm}$

Step 4: Find $AG$, the diagonal of the cuboid

Now, $AG$ is the diagonal of the cuboid, and it can be calculated as the hypotenuse of the triangle $\triangle ACG$, where:

• $AC = 29.23 \, \text{cm}$ (diagonal of the base).
• Height $CG = AB = 85.84 \, \text{cm}$.

Using the Pythagorean theorem again: $AG = \sqrt{AC^2 + CG^2}$ Substitute: $AG = \sqrt{(29.23)^2 + (85.84)^2}$ Calculate: $AG = \sqrt{854.32 + 7370.71} = \sqrt{8225.03}$ $AG \approx 90.68 \, \text{cm}$

Final Answer:

The length of $AG$ is approximately: $\boxed{90.68 \, \text{cm}}$

Would you like more details on any of the steps, or have additional questions?

Related Questions:

1. How would the solution change if the angle $\angle DAB$ were $60^\circ$ instead of $75^\circ$?
2. What is the formula for the diagonal of a cuboid in terms of its dimensions?
3. How is $AC$, the diagonal of the base, derived geometrically in cuboids?
4. What is the significance of the trigonometric function $\tan$ in finding $AB$?
5. Can you generalize the calculation for any cuboid when all three dimensions are known?

Tip:

For 3D geometry problems, always break down the diagonal into smaller triangles to apply the Pythagorean theorem step by step.