An emergency water sprinkler projects water across an angle of 87 °. It is installed in the roof at a height of 8 m, and the corridor beneath slopes at an angle of 17.6 °. What length of ground is sprayed by this sprinkler?

We are tasked with finding the length of ground that is sprayed by a sprinkler, given that it projects water over an angle of 87°, from a height of 8 meters, with the corridor beneath it sloping at an angle of 17.6°.

To approach this problem:

### Step 1: Set up the triangle
- The sprinkler sprays over a horizontal angle of 87°, but this is not directly useful because the ground slopes. We can instead treat this as a trigonometric problem.
- The sprinkler sprays water downwards, creating a triangle formed by the sprinkler, the start of the sprayed water, and the end of the sprayed area on the slope.

### Step 2: Identify the angles involved
- The sprinkler is at an angle of \( 87^\circ / 2 = 43.5^\circ \) to the left and right of its vertical axis (since the spray covers an equal area on both sides).
- The slope of the corridor adds a complication, as we need to consider the effective angle at which the water hits the sloped ground. The slope angle is 17.6°, so we need to account for how the water projects in relation to the slope.

### Step 3: Use trigonometry to calculate the length of ground sprayed
- We can think of the problem in terms of a triangle where the height is 8 meters, and we need to find the horizontal distance along the slope.
- The effective angle at which the water hits the ground can be considered as the sum of the spray angle and the slope angle.

Let me calculate the horizontal distance using trigonometric relations.The length of the ground that is sprayed by the sprinkler, considering the slope of the corridor, is approximately 4.42 meters.

Would you like a more detailed breakdown of the solution, or do you have any further questions?

Here are some related questions to deepen your understanding:
1. How would the length change if the slope angle increased?
2. What happens to the ground length if the spray angle is reduced to 60°?
3. How would you calculate the horizontal distance if the ground were flat?
4. If the height of the sprinkler were doubled, what would the new length be?
5. How does the angle of elevation of the sprinkler affect the length of the sprayed area?

**Tip:** Always make sure to convert angles to radians when using trigonometric functions in most calculators and programming environments.

An emergency water sprinkler projects water across an angle of 87°. It is installed in the roof at a height of 8 m, and the corridor beneath slopes at an angle of 17.6°. What length of ground is sprayed by this sprinkler?

Solve a trigonometric problem involving an emergency water sprinkler projecting water across an angle of 87° from a height of 8 m onto a corridor sloping at 17.6°. This problem uses trigonometric principles and geometry to determine the length of the sprayed area. Suitable for Grades 10-12.

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