Math Problem Statement
A ground observer sights a weather balloon to the east at an angle of elevation of 15 degree... A second observer 2.35 km to the east of the first also sights the balloon to the east at an angle of elevation of 22 degree. How high is the balloon?
Solution
To solve this problem, we can use trigonometry, specifically the tangent function, since it relates the angle of elevation to the height and distance of an object.
We are given two observers and their angles of elevation to the weather balloon. The situation can be modeled as two right triangles, where the height of the balloon is the common side and the horizontal distances are the bases of the triangles.
Let:
- be the height of the balloon (what we need to find).
- be the distance from the first observer to the point directly below the balloon.
Information given:
- The first observer has an angle of elevation of 15°.
- The second observer is 2.35 km (or 2350 m) to the east of the first observer and has an angle of elevation of 22°.
Step 1: Using the tangent function for both observers
For the first observer:
This gives us:
For the second observer:
This gives us:
Step 2: Set the two expressions for equal to each other
Since both expressions represent the height of the balloon:
Step 3: Solve for
We can now solve for by expanding and rearranging the equation:
Step 4: Calculate the value of
Let me calculate .It seems that the value of came out negative, which suggests there may have been an issue with the trigonometric calculation. Let me recheck the setup.
Step 5: Rethink the trigonometric model
We know that:
and
Let's reattempt solving it with a focus on recalculating the final expressions properly.It seems like I can’t do more advanced data analysis right now. Please try again later.
Would you like to see a detailed breakdown of how to proceed manually, or try something else? Feel free to ask any questions.
Here are 5 related questions to explore:
- What other trigonometric functions can help solve height and distance problems?
- How can angle differences between observers affect the calculation?
- What methods are useful for checking the consistency of trigonometric equations?
- How would the calculation change if one observer was farther away?
- What are some real-world applications of trigonometry beyond balloon sightings?
Tip: Always double-check the units (e.g., meters vs. kilometers) when solving geometry problems.
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Math Problem Analysis
Mathematical Concepts
Trigonometry
Right Triangle Geometry
Tangent Function
Formulas
tan(θ) = opposite / adjacent
h = x * tan(15°)
h = (x + 2350) * tan(22°)
Theorems
Tangent Ratio in Right Triangles
Suitable Grade Level
Grades 10-12
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