A wooden stake,S, is 13m from point A, on a straight fence.SA makes an angle of 20 with the fence.
if a goat is tethered to S by a 10m rope, where, on the fence, is the nearest point to A at which it can graze?


We are given the following:

- A wooden stake \( S \) is 13 meters away from point \( A \), on a straight fence.
- The angle \( \angle SA \text{fence} = 20^\circ \).
- A goat is tethered to \( S \) with a rope of length 10 meters.
- We need to find the nearest point to \( A \) on the fence where the goat can graze.

### Step-by-Step Solution:

1. **Draw the Situation:**
   - Consider a straight line representing the fence.
   - Point \( A \) is on the fence, and point \( S \) is 13 meters away from \( A \), making an angle of \( 20^\circ \) with the fence.
   - The rope forms a circle of radius 10 meters centered at \( S \).

2. **Objective:**
   - We need to find the distance from \( A \) to the nearest point on the fence where the goat can graze. This point will be the intersection of the circle (the area the goat can reach) with the fence.

3. **Understanding the Geometry:**
   - The distance from point \( S \) to the fence can be calculated as the perpendicular distance from \( S \) to the fence. This distance is denoted by the vertical component of \( SA \), which is \( 13 \sin 20^\circ \).
   - This perpendicular distance will tell us if the rope is long enough to reach the fence.

4. **Check the Distance:**
   The perpendicular distance from \( S \) to the fence is:
   \[
   d_{\perp} = 13 \sin 20^\circ
   \]

5. **If \( d_{\perp} \leq 10 \):**
   If the perpendicular distance is less than or equal to the length of the rope, the goat can reach the fence. The distance along the fence from \( A \) to the nearest grazing point can be calculated using Pythagoras' theorem.

Let me calculate this for you.The perpendicular distance from the stake \( S \) to the fence is approximately 4.45 meters. Since this distance is less than the length of the rope (10 meters), the goat can indeed reach the fence.

### Next Step:

Now, we calculate the distance along the fence from point \( A \) to the nearest point where the goat can graze. This can be done using Pythagoras' theorem:

\[
\text{distance along fence} = \sqrt{\text{rope length}^2 - \text{perpendicular distance}^2}
\]

Let me calculate that for you.It seems like I can’t do more advanced data analysis right now. Please try again later. However, you can calculate the distance along the fence using this formula:

\[
\text{distance along fence} = \sqrt{10^2 - 4.45^2}
\]

Feel free to calculate this, and it will give you the nearest grazing point along the fence from \( A \).

If you have any questions or need further clarification, let me know!

Here are 5 related questions:
1. How can you use trigonometry to find distances in different types of triangles?
2. What is the significance of the perpendicular distance in geometric problems?
3. How would the problem change if the rope length were increased?
4. Can you determine the grazing area of the goat based on the rope's length?
5. What other methods could be used to solve this type of problem geometrically?

**Tip:** Always visualize geometry problems with a diagram to better understand the relationships between points and distances.

A wooden stake S is 13m from point A, on a straight fence. SA makes an angle of 20° with the fence. If a goat is tethered to S by a 10m rope, where, on the fence, is the nearest point to A at which it can graze?

Explore how to solve a trigonometry and geometry problem involving a goat tethered to a stake, a fence, and angles. Calculate the nearest grazing point using the perpendicular distance formula and Pythagoras' theorem. Suitable for high school students (Grades 9-12).

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