SOLUTION: PROBLEM SOLVING. Analyze. Solve. Answer. Draw diagrams wherever necessary. Round off your final answers up to 2 decimal places. Law of Sines 20 pts Problem # 1. A scarecrow

Question 1193190: PROBLEM SOLVING. Analyze. Solve. Answer.
Draw diagrams wherever necessary. Round off your final answers up to 2 decimal places.
Law of Sines
20 pts Problem # 1. A scarecrow was built in the middle of a cornfield. A naughty crow and seemingly brave crow landed on top of the scarecrow and upon being seen by the farmer, the crow was chased away. As a result, the scarecrow is slightly tilted toward the sun at an angle of 12° from the vertical and casts a 3-meter shadow. The angle of elevation from the tip of the shadow to the top of the scarecrow is 27°.
• Draw an appropriate diagram for the problem.
• Determine the length of the scarecrow.
• Suppose more crows kept landing on the top of the scarecrow, gradually tilting it further down. Assuming that the angle of elevation from the tip of the shadow to the top of the scarecrow didn't change. What is the angle of tilt of the scarecrow from the vertical if the scarecrow casts a shadow of:
2 meters?
1 meter?
Found 2 solutions by yurtman, ikleyn:
Answer by yurtman(42)   (Show Source): You can put this solution on YOUR website!
Certainly, let's analyze, solve, and answer the problem about the tilted scarecrow.
**1. Draw a Diagram:**
[Image of a diagram showing a tilted scarecrow casting a shadow.
* The scarecrow is tilted at an angle of 12° from the vertical.
* The shadow of the scarecrow is 3 meters long.
* The angle of elevation from the tip of the shadow to the top of the scarecrow is 27°.
* Label the height of the scarecrow as 'h', the length of the shadow as 's', and the angle of tilt as 'θ'.]
**2. Determine the Length of the Scarecrow:**
* **Analyze:**
* We have a right triangle formed by the scarecrow, its shadow, and the ground.
* We know the angle of elevation (27°) and the length of the shadow (3 meters).
* We need to find the height of the scarecrow (h).
* **Solve:**
* Using the tangent function in trigonometry:
* tan(27°) = h / 3
* h = 3 * tan(27°)
* h ≈ 1.52 meters
* **Answer:** The length of the scarecrow is approximately 1.52 meters.
**3. Determine the Angle of Tilt for Different Shadow Lengths:**
* **Analyze:**
* The angle of elevation remains constant at 27°.
* We need to find the angle of tilt (θ) for different shadow lengths.
* **Solve:**
* **For a shadow length of 2 meters:**
* tan(27°) = h / 2
* h = 2 * tan(27°)
* h ≈ 1.01 meters
* Now, we have a right triangle with the height (h) and the angle of elevation (27°).
* We can use the tangent function to find the angle of tilt (θ):
* tan(θ) = h / 3
* tan(θ) = 1.01 / 3
* θ = arctan(1.01 / 3)
* θ ≈ 18.58°
* **For a shadow length of 1 meter:**
* tan(27°) = h / 1
* h = 1 * tan(27°)
* h ≈ 0.51 meters
* tan(θ) = h / 3
* tan(θ) = 0.51 / 3
* θ = arctan(0.51 / 3)
* θ ≈ 9.61°
* **Answer:**
* If the shadow length is 2 meters, the angle of tilt is approximately 18.58°.
* If the shadow length is 1 meter, the angle of tilt is approximately 9.61°.
**Note:** These calculations assume that the scarecrow remains rigid and only tilts at its base.
I hope this helps! Let me know if you have any other questions.

Answer by ikleyn(52835)   (Show Source): You can put this solution on YOUR website!
.
Problem # 1. A scarecrow was built in the middle of a cornfield.
A naughty crow and seemingly brave crow landed on top of the scarecrow and upon being seen by the farmer,
the crow was chased away. As a result, the scarecrow is slightly tilted toward the sun
at an angle of 12° from the vertical and casts a 3-meter shadow. The angle of elevation
from the tip of the shadow to the top of the scarecrow is 27°.
Determine the length of the scarecrow.
~~~~~~~~~~~~~~~~~~~~~~~~~~

        The problem in the post by tutor @yurtman is solved incorrectly.
        I came to bring a correct solution.

        I will solve here first part, only, for determining the length of the scarecrow.

We have a triangle ABC. Vertex A is the base of the scarecrow. Vertex B is the tip of the shadow on the ground. The side AB is horizontal on the ground. It represents the shadow on the ground and its length is 3 meters. Point C is the top of the scarecrow. Angle B is 27° : it is the angle of elevation from the tip of the shadow to the top of the scarecrow. Side AC makes the angle of 12° with vertical toward the sun; so the angle A is 90° + 12° = 102°. They want we find the length AC, which is the length of the scarecrow. First, calculate the angle C of the triangle as the third angle of the triangle ABC m(C) = 180° - 102° - 27° = 51°. Now write the sine law equation = = AC = = = 1.753 meters. ANSWER. The length of the scarecrow is 1.753 meters.

Solved.

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