Math Problem Statement

The shape below is a cuboid. Find the length of AG to 2 d.p.

Solution

To solve for the length of AGAG, we can follow these steps based on the diagram and the provided information:

Given:

  • The cuboid has a base with a length of 42mm42 \, \text{mm} (the length of ADAD).
  • The angle at AA, DAB=36\angle DAB = 36^\circ and BAH=27\angle BAH = 27^\circ, are provided.
  • AGAG is the diagonal from point AA to point GG, which is the corner opposite to AA 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 (ABAB)

Using the angle DAB=36\angle DAB = 36^\circ and the adjacent side AD=42mmAD = 42 \, \text{mm}, we can use the tangent function to find ABAB.

tan(36)=ABAD\tan(36^\circ) = \frac{AB}{AD} AB=AD×tan(36)AB = AD \times \tan(36^\circ) AB=42×tan(36)AB = 42 \times \tan(36^\circ) AB42×0.7265=30.51mmAB \approx 42 \times 0.7265 = 30.51 \, \text{mm}

Step 2: Find the height of the cuboid (BHBH)

Now, using the angle BAH=27\angle BAH = 27^\circ and the adjacent side ABAB, we can calculate BHBH.

tan(27)=BHAB\tan(27^\circ) = \frac{BH}{AB} BH=AB×tan(27)BH = AB \times \tan(27^\circ) BH30.51×0.5095=15.54mmBH \approx 30.51 \times 0.5095 = 15.54 \, \text{mm}

Step 3: Find the diagonal AGAG

Now that we have the length of ADAD, ABAB, and BHBH, we can use the Pythagorean theorem to find the length of AGAG.

First, calculate the diagonal AHAH of the face ABEHABEH: AH2=AB2+BH2AH^2 = AB^2 + BH^2 AH2=(30.51)2+(15.54)2AH^2 = (30.51)^2 + (15.54)^2 AH2=930.60+241.47=1172.07AH^2 = 930.60 + 241.47 = 1172.07 AH=1172.0734.24mmAH = \sqrt{1172.07} \approx 34.24 \, \text{mm}

Finally, use the Pythagorean theorem again to find AGAG, where AGAG is the diagonal of the cuboid: AG2=AH2+AD2AG^2 = AH^2 + AD^2 AG2=(34.24)2+(42)2AG^2 = (34.24)^2 + (42)^2 AG2=1172.07+1764=2936.07AG^2 = 1172.07 + 1764 = 2936.07 AG=2936.0754.19mmAG = \sqrt{2936.07} \approx 54.19 \, \text{mm}

Final Answer:

The length of AGAG 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:

  1. How do you calculate diagonal lengths in a cuboid?
  2. Can you explain the use of trigonometry in this solution?
  3. What is the Pythagorean theorem, and how is it applied here?
  4. How do you interpret angles in 3D geometry problems like this?
  5. 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