Math Problem Statement
The angle of elevation to a nearby tree from a point on the ground is measured to be 45, degrees45 ∘ . How tall is the tree if the point on the ground is 8080 feet from the bottom of the tree? Round your answer to the nearest tenth of a foot if necessary. x 80 Answer
Solution
We can solve this problem using trigonometry. Specifically, we will use the tangent function, which is defined as:
Here,
- is the angle of elevation,
- the opposite side is the height of the tree, which we will call ,
- the adjacent side is the horizontal distance from the point on the ground to the base of the tree, which is 80 feet.
The formula for this scenario becomes:
We know that , so the equation simplifies to:
Multiplying both sides by 80:
So, the height of the tree is 80 feet.
Would you like more details on this problem, or do you have any questions?
Here are some related questions to explore:
- How would the height change if the angle of elevation were 30° instead of 45°?
- What if the distance from the point on the ground to the base of the tree were doubled?
- How could you apply this concept to find the height of a building?
- What are the other trigonometric functions, and how do they relate to right triangles?
- How can you measure angles of elevation in real life using a protractor or smartphone?
Tip: In right triangle problems like this one, always check that your angle is in the correct quadrant and remember that for a 45° angle in a right triangle, the opposite and adjacent sides will be equal in length.
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Math Problem Analysis
Mathematical Concepts
Trigonometry
Right Triangles
Tangent Function
Formulas
tan(θ) = opposite / adjacent
Theorems
Tangent function and its application in right triangles
Suitable Grade Level
Grades 7-9
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