Observatories A,B

The target C is observed from two artillery observatories, A and B, 296 m apart. At the same time, angle BAC = 52°42" and angle ABC = 44°56". Calculate the distance of the target C from observatory A.

Correct answer:

AC =  210.9291 m

Step-by-step explanation:


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Dr. Math
To determine the distance of target C from observatory A , we can use the Law of Sines in triangle ABC . Here's the step-by-step solution:

Step 1:

Understand the Given Information
- The distance between observatories A and B is AB = 296 m .
- Angle BAC = 52°42' .
- Angle ABC = 44°56' .

We need to find the distance AC , which is the distance from observatory A to target C .

Step 2:

Convert Angles to Decimal Degrees
First, convert the angles from degrees, minutes, and seconds to decimal degrees for easier calculations.

1. Angle BAC = 52°42' :
  - 42' = 4260 = 0.7° .
  - So, BAC = 52 + 0.7 = 52.7° .

2. Angle ABC = 44°56' :
  - 56' = 5660 ≈ 0.9333° .
  - So, ABC = 44 + 0.9333 = 44.9333° .

Step 3:

Find Angle ACB
In any triangle, the sum of the angles is 180° . Therefore:
ACB = 180° - BAC - ABC

ACB = 180° - 52.7° - 44.9333°

ACB = 82.3667°

Step 4:

Apply the Law of Sines
The Law of Sines states:
ACsin(ABC) = ABsin(ACB)


We are solving for AC , so rearrange the formula:
AC = AB · sin(ABC)sin(ACB)


Substitute the known values:
AC = 296 · sin(44.9333°)sin(82.3667°)

Step 5:

Calculate the Sine Values
Using a calculator:
- sin(44.9333°) ≈ 0.7068 .
- sin(82.3667°) ≈ 0.9911 .

Now substitute these values into the equation:
AC = 296 · 0.70680.9911

AC ≈ 209.21280.9911

AC ≈ 211.1 m

Final Answer:


The distance of the target C from observatory A is approximately:

211.1 m







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