Atiba wants to measure the width of the Squamscott River. Standing under a tree I on the river bank, Atiba sights a rock at the nearest point R on the opposite bank. Then Atiba walks to a point P on the river bank that is 50.0 meters from T, and makes RTP a right angle. Atiba then measures RPT and obtains 76.8 degrees. How wide is the river?

This is a classic trigonometry problem involving a right triangle. Here's a breakdown of how to solve it:

### Step-by-step solution:

1. **Identify the triangle and angles:**
   - Let the width of the river be \( x \). This is the distance from point \( R \) (the nearest point on the opposite bank) to point \( T \) (where Atiba initially stands under the tree).
   - Atiba walks 50 meters from \( T \) to \( P \), which creates a right triangle \( RTP \), with the right angle at \( T \).
   - The angle \( \angle RPT \) is measured as 76.8 degrees.

2. **Apply trigonometry:**
   The tangent of an angle in a right triangle is given by:
   \[
   \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
   \]
   In this case:
   \[
   \tan(76.8^\circ) = \frac{x}{50}
   \]
   where \( x \) is the width of the river, and 50 meters is the distance from \( T \) to \( P \).

3. **Solve for \( x \):**
   Rearranging the equation to solve for \( x \):
   \[
   x = 50 \times \tan(76.8^\circ)
   \]
   Now we calculate this value.

I'll calculate that for you.The width of the Squamscott River is approximately 213.18 meters.

Would you like further details or clarifications on any part of the solution? Here are 5 related questions to explore further:

1. How would the solution change if Atiba measured a different angle?
2. Can you use the sine or cosine functions to solve this problem?
3. What happens if Atiba walked a different distance instead of 50 meters?
4. How would you verify the accuracy of the measurement in a real-world scenario?
5. How could this method be used to measure other inaccessible distances?

**Tip:** Always double-check your angle measurements in trigonometry problems to ensure accurate results.

Atiba wants to measure the width of the Squamscott River. Standing under a tree T on the river bank, Atiba sights a rock at the nearest point R on the opposite bank. Then Atiba walks to a point P on the river bank that is 50.0 meters from T, and makes RTP a right angle. Atiba then measures RPT and obtains 76.8 degrees. How wide is the river?

Learn how to calculate the width of the Squamscott River using trigonometry. This step-by-step solution uses the tangent function in a right-angled triangle to find the river's width, given an angle of 76.8 degrees and a distance of 50 meters. Suitable for students in Grades 9-12.

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