Math Problem Statement
Solution
To solve for the length of , we can follow these steps based on the diagram and the provided information:
Given:
- The cuboid has a base with a length of (the length of ).
- The angle at , and , are provided.
- is the diagonal from point to point , which is the corner opposite to in the cuboid.
We can break down the problem using trigonometry and the dimensions of the cuboid:
Step 1: Find the height of the cuboid ()
Using the angle and the adjacent side , we can use the tangent function to find .
Step 2: Find the height of the cuboid ()
Now, using the angle and the adjacent side , we can calculate .
Step 3: Find the diagonal
Now that we have the length of , , and , we can use the Pythagorean theorem to find the length of .
First, calculate the diagonal of the face :
Finally, use the Pythagorean theorem again to find , where is the diagonal of the cuboid:
Final Answer:
The length of is approximately 54.19 mm, rounded to two decimal places.
Would you like a further breakdown or clarification on any step? Here are some related questions:
- How do you calculate diagonal lengths in a cuboid?
- Can you explain the use of trigonometry in this solution?
- What is the Pythagorean theorem, and how is it applied here?
- How do you interpret angles in 3D geometry problems like this?
- Can you generalize this method for any cuboid with different dimensions?
Tip: When solving geometry problems involving angles and dimensions, always double-check if you need to apply the trigonometric functions or the Pythagorean theorem step by step for accuracy.
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Math Problem Analysis
Mathematical Concepts
Trigonometry
3D Geometry
Pythagoras' Theorem
Formulas
tan(θ) = opposite/adjacent
Pythagorean theorem: a² + b² = c²
Theorems
Pythagorean Theorem in 3D space
Suitable Grade Level
Grades 9-12