A smokestack is 160 feet high. A guy wire must be fastened to the stack 30 feet from the top. The guy wire makes an angle of 40 with the ground. Find the length of the guy wire rounded to the nearest foot.

We can solve this problem using trigonometry. Here's the step-by-step process:

1. **Identify the key parts of the problem:**
   - Height of the smokestack: \(160 \, \text{feet}\)
   - The guy wire is fastened 30 feet from the top, so the height from the ground to where the wire is attached is:
     \[
     h = 160 - 30 = 130 \, \text{feet}
     \]
   - Angle the guy wire makes with the ground: \(40^\circ\)
   - We are asked to find the length of the guy wire, which we'll call \(L\).

2. **Set up the right triangle:**
   - The height from the ground to where the wire is fastened is one leg of the right triangle: \(130 \, \text{feet}\).
   - The length of the guy wire \(L\) is the hypotenuse of the triangle.
   - The angle between the guy wire and the ground is \(40^\circ\).

3. **Use the sine function:**
   The sine function relates the opposite side (height) to the hypotenuse (guy wire length):
   \[
   \sin(40^\circ) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{130}{L}
   \]
   
4. **Solve for \(L\):**
   \[
   L = \frac{130}{\sin(40^\circ)}
   \]

Now, I'll calculate the length \(L\).The length of the guy wire is approximately **202 feet**.

Do you want more details or have any questions about the solution?

Here are 5 related questions for practice:
1. What would be the length of the guy wire if the angle with the ground were \(50^\circ\)?
2. If the smokestack height were increased to 180 feet, how would the length of the wire change?
3. What is the horizontal distance between the base of the smokestack and the point where the guy wire is anchored to the ground?
4. How would you solve this problem if the angle were not given but the length of the guy wire was known?
5. What is the minimum angle the guy wire can make with the ground to ensure it’s 202 feet long?

**Tip:** Trigonometric functions like sine and cosine are extremely useful in solving problems involving right triangles and angles.

A smokestack is 160 feet high. A guy wire must be fastened to the stack 30 feet from the top. The guy wire makes an angle of 40° with the ground. Find the length of the guy wire rounded to the nearest foot.

Learn how to calculate the length of a guy wire fastened to a smokestack, given the height of the smokestack and the angle the wire makes with the ground. This problem involves using trigonometric ratios, specifically the sine function. Suitable for students in Grades 10-12.

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