A Student Is Standing 20 Feet Away From The Base Of A Tree. He Looks To The Top Of The Tree At A 50° Angle Of Elevation. His Eyes Are 5 Feet Above The Ground. Using Cos 50° 0.64, What Is The Height Of The Tree To The Nearest Foot? 350 36 Feet 24 Feet 29 Feet 31 Feet
The correct answer and explanation is:
The correct answer is 29 feet.
Explanation:
We can solve this problem using trigonometry, specifically the tangent function, since we are dealing with a right triangle formed by the student, the tree, and the line of sight.
Step 1: Define the Known Values
- The student is 20 feet away from the base of the tree.
- The angle of elevation to the top of the tree is 50°.
- The student’s eye level is 5 feet above the ground.
Step 2: Set Up the Trigonometric Equation
We let h be the height of the tree above the student’s eye level. The tangent function is used because we have the opposite and adjacent sides in the right triangle: tan(50∘)=oppositeadjacent\tan(50^\circ) = \frac{\text{opposite}}{\text{adjacent}} tan(50∘)=h20\tan(50^\circ) = \frac{h}{20}
From a calculator: tan(50∘)≈1.19\tan(50^\circ) \approx 1.19 1.19=h201.19 = \frac{h}{20}
Step 3: Solve for h
Multiply both sides by 20: h=20×1.19h = 20 \times 1.19 h=23.8≈24 feeth = 23.8 \approx 24 \text{ feet}
Step 4: Add the Student’s Eye Height
The total height of the tree is: 24+5=29 feet24 + 5 = 29 \text{ feet}
Thus, the correct answer is 29 feet.
Generated Image Description:
A student is standing on the ground, 20 feet away from a tall tree. A right triangle is formed by the student’s eye level (5 feet above the ground), the top of the tree, and the base of the tree. A dotted line represents the student’s line of sight at a 50° angle to the top of the tree. The height of the tree is labeled as 29 feet.
Here is the generated image illustrating the problem. Let me know if you need any modifications or explanations!
