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:

tan(θ)=oppositeadjacent\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}

Here,

  • θ=45\theta = 45^\circ is the angle of elevation,
  • the opposite side is the height of the tree, which we will call hh,
  • 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:

tan(45)=h80\tan(45^\circ) = \frac{h}{80}

We know that tan(45)=1\tan(45^\circ) = 1, so the equation simplifies to:

1=h801 = \frac{h}{80}

Multiplying both sides by 80:

h=80h = 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:

  1. How would the height change if the angle of elevation were 30° instead of 45°?
  2. What if the distance from the point on the ground to the base of the tree were doubled?
  3. How could you apply this concept to find the height of a building?
  4. What are the other trigonometric functions, and how do they relate to right triangles?
  5. 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