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:
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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:
- 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 .
First, convert the angles from degrees, minutes, and seconds to decimal degrees for easier calculations.
1. Angle BAC = 52°42' :
- 42' = 42/60 = 0.7° .
- So, BAC = 52 + 0.7 = 52.7° .
2. Angle ABC = 44°56' :
- 56' = 56/60 ≈ 0.9333° .
- So, ABC = 44 + 0.9333 = 44.9333° .
In any triangle, the sum of the angles is 180° . Therefore:
The Law of Sines states:
We are solving for AC , so rearrange the formula:
Substitute the known values:
Using a calculator:
- sin(44.9333°) ≈ 0.7068 .
- sin(82.3667°) ≈ 0.9911 .
Now substitute these values into the equation:
The distance of the target C from observatory A is approximately:
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 DegreesFirst, convert the angles from degrees, minutes, and seconds to decimal degrees for easier calculations.
1. Angle BAC = 52°42' :
- 42' = 42/60 = 0.7° .
- So, BAC = 52 + 0.7 = 52.7° .
2. Angle ABC = 44°56' :
- 56' = 56/60 ≈ 0.9333° .
- So, ABC = 44 + 0.9333 = 44.9333° .
Step 3:
Find Angle ACBIn 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 SinesThe Law of Sines states:
AC/sin(ABC) = AB/sin(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 ValuesUsing a calculator:
- sin(44.9333°) ≈ 0.7068 .
- sin(82.3667°) ≈ 0.9911 .
Now substitute these values into the equation:
AC = 296 · 0.7068/0.9911
AC ≈ 209.2128/0.9911
AC ≈ 211.1 m
Final Answer:
The distance of the target C from observatory A is approximately:
211.1 m
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You need to know the following knowledge to solve this word math problem:
- planimetrics
- triangle
- The Law of Cosines
- The Law of Sines
- basic functions
- percentages
- numbers
- fractions
- goniometry and trigonometry
- sine
Units of physical quantities:
Grade of the word problem:
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