Math Problem Statement

Calculate the bearing of C from A given the positions of three tents, A, B, and C. B is 5 m due south of tent A. C is 7.5 m due west of tent B.

Solution

The diagram you provided shows three tents AA, BB, and CC arranged in a right-angled triangle. We are tasked with calculating the bearing of point CC from point AA.

The information given is:

  • Tent BB is 5 m due south of AA (vertical leg of the triangle).
  • Tent CC is 7.5 m due west of BB (horizontal leg of the triangle).

Since the right angle is at BB, we need to find the bearing of CC from AA. Bearings are measured clockwise from the north direction. In this case, we need to find the angle θ\theta that tent CC makes with respect to the north direction from AA.

Steps to Solve:

  1. Determine the angle θ\theta using trigonometry: The triangle formed by AA, BB, and CC is a right triangle. The angle θ\theta is the angle between the line ACAC and the vertical leg ABAB. We can use the tangent function: tan(θ)=oppositeadjacent=7.55\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{7.5}{5} So, θ=tan1(7.55)\theta = \tan^{-1}\left(\frac{7.5}{5}\right)

  2. Calculate the value of θ\theta: θ=tan1(1.5)56.31\theta = \tan^{-1}(1.5) \approx 56.31^\circ

  3. Calculate the bearing: Bearings are measured clockwise from the north, so we need to add 180180^\circ to the calculated angle θ\theta, because the direction of CC from AA is in the southwest quadrant. Thus, the bearing of CC from AA is: 180+56.31=236.3180^\circ + 56.31^\circ = 236.3^\circ

Final Answer:

The bearing of CC from AA is approximately 236.3236.3^\circ.


Would you like further details or explanations?

Here are 5 related questions to explore:

  1. How do you calculate a bearing if you are given coordinates instead of distances?
  2. What other trigonometric functions could be useful in finding angles in triangles?
  3. How would you calculate the distance between AA and CC?
  4. What is the difference between true bearing and relative bearing?
  5. Can bearings ever be negative, and if so, how are they handled?

Tip: Always ensure you know whether bearings are being measured clockwise or counterclockwise, as this can impact the final answer significantly.

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Bearings
Right-Angle Triangles

Formulas

tan(θ) = opposite / adjacent
θ = tan^(-1)(opposite / adjacent)
Bearing = 180° + θ

Theorems

Trigonometric Ratio (Tangent)

Suitable Grade Level

Grades 8-10